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DYNAMICS AND NUMERICS OF
GENERALISED EULER EQUATIONS
by
Xingyou (Philip) ZHANG
A Thesis Submitted to Massey University
In Partial Fulfillment of the Requirements
For the Degree of Ph.D in Mathematics
Palmerston North, New Zealand
2008
 
Xingyou (Philip) Zhang 2008
Abstract
This thesis is concerned with the well-posedness, dynamical properties and
numerical treatment of the generalised Euler equations on the Bott-Virasoro
group with respect to the general Hk metric , k ≥ 2.
The term “generalised Euler equations” is used to describe geodesic equa-
tions on Lie groups, which unifies many differential equations and has found
many applications in such as hydrodynamics, medical imaging in the compu-
tational anatomy, and many other fields. The generalised Euler equations on
the Bott-Virasoro group for k = 0, 1 are well-known and intensively studied—
the Korteweg-de Vries equation for k = 0 and the Camassa-Holm equation
for k = 1. Unlike these, the equations for k ≥ 2, which we call the mod-
ified Camassa-Holm (mCH) equation, is not known to be integrable. This
distinction motivates the study of the mCH equation.
In this thesis, we derive the mCH equation and establish the short time
existence of solutions, the well-posedness of the mCH equation, long time
existence, the existence of the weak solutions, both on the circle S and R, and
three conservation laws, show some quite interesting properties, for example,
they do not lead to the blowup in finite time, unlike the Camassa-Holm
equation.
We then consider two numerical methods for the modified Camassa-Holm
equation: the particle method and the box scheme. We prove the convergence
result of the particle method. The numerical simulations indicate another
interesting phenomenon: although mCH does not admit blowup in finite
time, it admits solutions that blow up (which means their maximum value
becomes infinity) at infinite time, which we call weak blowup. We study
this novel phenomenon using the method of matched asymptotic expansion.
A whole family of self-consistent blowup profiles is obtained. We propose a
mechanism by which the actual profile is selected that is consistent with the
simulations, but the mechanism is only partly supported by the analysis.
We study the four particle systems for the mCH equation finding numeri-
cal evidence both for the non-integrability of the mCH equations and for the
existence of the fourth integral. We also study the higher dimensional case
i
ABSTRACT ABSTRACT
and obtain the short time existence and well-posedness for the generalised
Euler equation in the two dimension case.
ii
Acknowledgements
Firstly, I would like to express my sincere thanks to my main supervisor,
Prof. Robert McLachlan, who has spent many, many hours over the last
four years enthusiastically and patiently teaching me the theory of dynami-
cal systems, geometric integration, how to implement numerically the various
mathematics ideas, and how to improve my English! I have always appreci-
ated your friendly manner and encouragement to try new ideas and attend
international conferences.
Special thanks also go to my supervisor Dr. Matt Perlmutter for your
encouragement, your investment of time and patiently explaining to me when
your were bombed with my geometry questions which are quite simple and
may be even stupid to you.
I would like to acknowledge all my PhD fellows, especially Dion and
Brett, for making my PhD experience at Massey much more rich and pleas-
ant. Many thanks to the IFS staff at Massey who are friendly and helpful,
especially to Kee Teo, for helping me to navigate my tutorship and other
intricacies of life at Massey.
I am grateful to acknowledge that this work has been supported by NZ-
IMA thematic PhD scholarship. Additional travel support was provided by
Education Ministry of New Zealand (to enable me to visit Chinese Academy
of Sciences, Beijing), NZIMA travel funding and IFSGRF of Massey Univer-
sity (to enable me to visit Newton Institute of Cambridge University and to
attend SciCADE07 in France), for which I am thankful.
A special thanks to my family, especially my wife, Wendy, and my son,
William, for the support and freedom to pursue my interests. I dedicate this
thesis to the memory of my father, who, together with my mother, brought
me up under the very hard condition, kept encouraging me to study for my
interests but passed away two years ago.
Xingyou Philip Zhang, July 11, 2008
iii
Contents
Abstract i
Acknowledgements iii
List of Spaces vii
1 Introduction 1
1.1 Euler Fluid Equations: A Brief History . . . . . . . . . . . . . 2
1.2 Arnold’s Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Shallow Water Equations . . . . . . . . . . . . . . . . . 4
1.2.2 Abstract Euler-Poincare´ Equations . . . . . . . . . . . 5
1.3 Particle solutions . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Numerical Approaches . . . . . . . . . . . . . . . . . . . . . . 8
1.4.1 Particle Method . . . . . . . . . . . . . . . . . . . . . . 8
1.4.2 Box Scheme . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.3 Multi-symplectic Methods . . . . . . . . . . . . . . . . 9
1.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Thesis Preview . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Preliminary Tools 14
2.1 PDE Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.1 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . 15
2.1.2 Kato Theory . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Riemannian Geometry . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1 Lie Group and Adjoint Representation . . . . . . . . . 24
2.3.2 Co-adjoint Representation of a Lie Group . . . . . . . 26
2.3.3 Invariant Metrics of Lie Groups . . . . . . . . . . . . . 26
2.3.4 Applications to Hydrodynamics . . . . . . . . . . . . . 29
iv
CONTENTS CONTENTS
3 Well-posedness 30
3.1 Derivation of the Equations . . . . . . . . . . . . . . . . . . . 30
3.1.1 Bott-Virasoro Group . . . . . . . . . . . . . . . . . . . 30
3.1.2 Derivation of the Equations . . . . . . . . . . . . . . . 31
3.2 Local Well-posedness . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.1 Conservation Laws . . . . . . . . . . . . . . . . . . . . 37
3.3 Global Well-posedness . . . . . . . . . . . . . . . . . . . . . . 38
3.3.1 Extra Properties for a = 0 . . . . . . . . . . . . . . . . 40
3.4 Weak Solutions for a = 0 . . . . . . . . . . . . . . . . . . . . . 44
3.5 The Whole Real Line Case . . . . . . . . . . . . . . . . . . . . 49
3.6 Remarks on the Generalisations . . . . . . . . . . . . . . . . . 51
3.7 Conjugate Points and Beyond . . . . . . . . . . . . . . . . . . 51
3.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4 Numerics 57
4.1 Particle methods . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Box Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5 Asymptotics 69
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 Asymptotic PDE . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.3 Are Steady Solutions Stable? . . . . . . . . . . . . . . . . . . . 75
5.3.1 Upstream Boundary Conditions for (5.12) . . . . . . . 75
5.3.2 Around the General Steady Solution (5.8) . . . . . . . 79
5.3.3 Around the Limit Steady Solution (5.9) . . . . . . . . . 83
5.4 The Family of Steady Solutions . . . . . . . . . . . . . . . . . 85
5.5 Remarks on the Camassa-Holm Equation . . . . . . . . . . . . 92
5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6 Four Particle Systems 95
6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.2 Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . . . 96
6.3 How to Compute Them? . . . . . . . . . . . . . . . . . . . . . 96
6.4 Four Particle Systems . . . . . . . . . . . . . . . . . . . . . . . 98
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7 Higher Dimensional Case 105
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.2 Local Well-posedness . . . . . . . . . . . . . . . . . . . . . . . 105
v
CONTENTS CONTENTS
8 Future Work 112
8.1 Stability of the Asymptotic Solutions . . . . . . . . . . . . . . 112
8.2 Positivity of Solutions . . . . . . . . . . . . . . . . . . . . . . 114
8.3 Higher Dimensional Case . . . . . . . . . . . . . . . . . . . . . 114
A Properties of the Green’s Function 115
B Multi-symplectic Formulation 119
B.1 Multi-symplectic Geometry . . . . . . . . . . . . . . . . . . . 119
B.2 Formulation for mCH . . . . . . . . . . . . . . . . . . . . . . 129
Bibliography 133
vi
List of Spaces
Here is the list of various spaces in Chapter 1 − Chapter 7.
The notation Its meaning
B(X, Y ) The space of all bounded linear operators from X to Y
C([a, b], X) The set of all continuous functions from [a, b] to X
Cα(Ω) Ho¨lder spaces defined in Section 2.1
C1,c(Ω) The set of all continuous functions with compact supports in Ω
and continuous first order derivatives in Ω
Diff(S) The set of all diffeomorphisms from S to S preserving the orientation
Ds(S) The set of all Hs diffeomorphisms on S
D̂(S) Bott-Virasoro group defined in Section 3.1
G(X, 1, β) The set of quasi-m-accretive operators in X defined in Section 2.1.2
GL(V ) The set of all invertible linear operators from V to V
Hs(Rn) W s,2(Rn)
Hs(S) The s-th order Sobolev space W s,2(S)
H∞(S)
∞⋂
s=1
Hs(S)
Lp(Ω) The set of all measurable functions u with
∫
Ω |u|pdx <∞
L∞(Ω) The set of essentially bounded measurable functions on Ω
R1 The standard one dimensional Euclidean space
Rn The standard n dimensional Euclidean space
S The unit circle R1/2piZ
SO(n) The group of special orthogonal transforms in Rn+1
W k,p(Ω) Sobolev spaces defined in Section 2.1
vii
Chapter 1
Introduction
However sublime are the researches on fluids which we owe to
Messrs Bernoulli, Clairaut and d’Alembert, they flow so naturally
from my two general formulae that one cannot sufficiently admire
this accord of their profound meditations with the simplicity of the
principles from which I have drawn my two equations...
——L. Euler, 1752.
The Euler fluid equation for the perfect (ie, inviscid incompressible) fluid
has been one of the most attractive PDEs in the mathematical physics since
it appeared about 250 years ago. It can be derived from the conservation
of the mass and momentum of the fluid. However, V.I. Arnold [3] proposed
in 1966 a completely different perspective which says that the Euler fluid
equation can be viewed as the geodesic equation on some diffeomorphism
groups with respect to some invariant metric. Arnold’s approach has laid a
theoretical foundation to exploit a simple construction in Lie group to unify
various dynamical systems in mathematical physics. Now the term “gener-
alised Euler equations” (or Euler-Poincare´ equations) are used to describe
the general geodesic equations on Lie groups.
The Korteweg-de Vries (KdV) equation and the Camassa-Holm (CH)
equation are two examples of generalised Euler equations. They can be
viewed as the Euler equations on the Virasoro group D̂s(S) (or on the diffeo-
morphism group Diff(S) for the limiting case) with respect to the L2 metric
(ie H0 metric) and H1 metric respectively on its Lie algebra [64]. These
equations read
mt + umx + 2uxm = a∂3xu
with m = u for the KdV equation and m = (1− ∂2x)u for the Camassa-Holm
equation.
1
CHAPTER 1. INTRODUCTION 1.1. EULER FLUID EQUATIONS: A BRIEF HISTORY
My thesis mainly concerns the analytical and dynamical properties of
their generalised version (which we call the modified Camassa-Holm equa-
tion)
mt + umx + 2uxm = a∂3xu with m = (1− ∂2x + · · ·+ (−1)k∂2kx )u, k ∈ N,
(1.1)
with the following motivations:
 Mathematically, KdV and Camassa-Holm have some significant differ-
ences in dynamics. For example, the Camassa-Holm equation leads to
blowup in finite time for some initial values and admits smooth solu-
tions for some other initials while for KdV [111] we have the global
wellposedness for all smooth enough initial values. So it is natural to
ask how the dynamics of the generalised Euler equations depends on
the metric on the Lie algebra? Or more specifically, how do the so-
lutions of the generalised Euler equations on D̂s(S) corresponding to
the Hk metric, with k 6= 0, 1, on its Lie algebra behave dynamically?
We know that the KdV and Camassa-Holm equations are integrable
systems, then how about the general Hk metric case?
 Just as the Euler fluid equation has the point vortex solutions, the
KdV equation has the so-called soliton solutions, while CH admits the
peakon solutions. All these solutions are the so-called “particle solu-
tions”. The peakons in the Camassa-Holm equation corresponds to the
Dirac δ function in the momentum, just the same as the vortex in the
fluid dynamics. The study of peakons (and solitons) are closely related
to the integrability, what can we say about the δ solutions of the gen-
eralised Euler equations for the Hk metric while we do not know they
are integrable or not?
 D. Holm et al [53] discussed the applications of the generalised Eu-
ler equations in the computational anatomy and mentioned that a
smoother kernel than the inverse of I−4 is used there. Mathematically,
this means that we need to consider the dynamics of the generalised
Euler equations of Hk metric other than H1 as in the Camassa-Holm
equation.
1.1 Euler Fluid Equations: A Brief History
Leonhard Euler, arguably one of the three greatest mathematicians in the
history of human beings, published a number of major pieces of work [35]
2
CHAPTER 1. INTRODUCTION 1.2. ARNOLD’S VIEWPOINT
through the 1750s setting up the main formulae for the study of fluid dynam-
ics: the continuity equation, the Laplace velocity potential equation, and the
Euler equations for the motion of an inviscid compressible fluid. After Euler’s
work, Cauchy [20] described in 1823 the conservation of mass and angular
momentum by PDEs. Material symmetry and frame invariance were used
by Cauchy [20] and Poisson [95] to reduce the constitutive equations. The
dissipative effects of internal frictional forces were modeled mathematically
by Navier [90], Poisson [95] etc.
In modern fluid dynamics, the Euler equations are in general used to term
the equations that govern the motion of an incompressible, inviscid fluid:



ut + u · ∇u = −∇p in R2 or R3
divu = 0 in R2 or R3.
(1.2)
where p stands for the pressure, u the velocity. They correspond to the
Navier-Stokes equations with zero viscosity, although they are usually written
in the form of conservation laws and emphasize the fact that they directly
represent conservation of mass and momentum.
For 250 years, Euler’s equations have formed an essential part of the
bedrock of our understanding of fluid flow. They are still one of the most
fascinating PDEs. In the memorial issue of Bulletin of AMS, no.4, Vol. 44,
2007, devoted to the 300th anniversary of Euler’s birth, two of the six surveys
are about the Euler fluid equations.
1.2 Arnold’s Viewpoint
Arnold [3] in 1966 introduced a completely new viewpoint on the Euler equa-
tions: they are equivalent to the equation of geodesics on a diffeomorphism
group with an invariant metric! His method uses one simple construction in
Lie group to give a unified approach to a great variety of differential dynami-
cal systems, from the simple (Euler) equation of a rotating top (corresponding
to the group SO(3) with the metric 〈ω, Iω〉, here I = diag(I1, I2, I3)) to the
(Euler) hydrodynamics equations (corresponding to the group SDiff(M) with
the metric L2). This led to a totally new stage of development of the Euler
equations.
Ebin and Marsden [34] in 1970 used this viewpoint to prove the well-
posedness for the Euler equations and Navier-Stokes equations of an incom-
pressible fluid on a (possibly with boundary) Riemannian manifold (cor-
3
CHAPTER 1. INTRODUCTION 1.2. ARNOLD’S VIEWPOINT
responding the group SDiff(M) with the L2 metric), which has not been
bettered until nowadays.
A curious application [5, 63] of this theory is an explanation of why long-
term dynamical weather forecasts are not reliable: Arnold’s explicit estimates
related to curvatures of diffeomorphism groups show that the earth’s weather
is essentially unpredictable after two weeks as the error in the initial condition
grows by a factor of 105 for that period, that is, one loses 5 digits of accuracy.
Another application [5, 63] is related to the Sakharov-Zeldovich problem
on whether a neutron star can extinguish by “reshaping” and turning to
radiation the excessive magnetic energy.
Now the equations of geodesics on Lie groups are called generalized Euler
equations (or Euler-Poincare´ equations), which include the Euler equations
in fluid dynamics and many interesting partial differential equations in math-
ematical physics and other areas, for example:
 Landau-Lifshitz equations of micromagnetics
 Template matching equations used in image processing.
With this term, the well-known Camassa-Holm equation can be also cat-
egorized as an Euler equation.
1.2.1 Shallow Water Equations
In the study of shallow water waves, Camassa and Holm [18] derived in 1993
the following partial differential equation
(I − ∂2x)ut + 2∂xu · (I − ∂2x)u+ u · (∂x − ∂3x)u = 0, (1.3)
by an asymptotic expansion directly in the Hamiltonian and intensively stud-
ied its properties: its complete integrability, its bi-Hamiltonian structure, in-
finite conservation laws and the existence of peaked soliton solutions. For
these reasons, this PDE is called the Camassa-Holm equation and often
considered as one of the most fascinating PDEs in mathematical physics.
Since the birth of this equation, many people have contributed to the well-
posedness study on the whole real line R or on the unit circle S: to mention
a few, Arnold and Khesin [5], Constantin and McKean [25], McKean [84],
and the references therein. Local well-posedness for (1.3) was discussed by
Constantin [27], Constantin and Escher [28] for the initial data in Hs(S) with
s ≥ 4 and s ≥ 3 respectively, and by Misio lek [88] with s > 3/2. Local well-
posedness in the non-periodic case was proved for the initial data in Hs(R)
4
CHAPTER 1. INTRODUCTION 1.2. ARNOLD’S VIEWPOINT
with s > 3/2 by Li and Olver [70] and Rodriguez-Blanco [97]. Classical so-
lutions can become singular in finite time if the initial momentum (I − ∂2x)u
changes sign. It is worthwhile to mention that Xin and Zhang [108] proved
the global existence of the weak solution in the energy space H1(R) with-
out any sign conditions on the initial value, and the uniqueness of this weak
solution is obtained under some restrictions on the solution [109].
Khesin and Misio lek [64] proved that the Camassa-Holm equation is the
equation of the geodesic flow associated to H1(S) metric on the diffeo-group
Diff(S) of the circle, which is the Euler-Poincare´ equation by using the La-
grangian associated with the H1 metric for the fluid velocity, ie, the La-
grangian as a function of the fluid velocity which is given by the quadratic
form,
l(u) = 12
∫
(u2 + u2x)dx.
1.2.2 Abstract Euler-Poincare´ Equations
Let G be a Lie group and g its associated Lie algebra (identified with the
tangent space to G at the identity element), with Lie bracket denoted by
[ξ, η] for ξ, η ∈ g. Let l : g 7→ R be a given Lagrangian and L : TG 7→ R the
right invariant Lagrangian on G obtained by translating l from the identity
element to other points of G via the right action of G on TG. A basic result
of Euler-Poincare´ theory [50, 52] is that the Euler-Lagrange equations for L
on G are equivalent to the (right) Euler-Poincare´ equations for l on g :
d
dt
δl
δξ = −ad
∗
ξ
δl
δξ .
Here adξ : g 7→ g is the adjoint operator of the linear map given by the Lie
bracket η 7→ [ξ, η], ad∗ξ : g∗ 7→ g∗ is given by (ad∗ξ(µ), η) = (µ, [ξ, η]), where
(·, ·) is the pairing between g∗ and g. The Euler-Poincare´ equation can also
be written in the variational form δ
∫
ldt = 0 for all the variations of the
form δξ = η˙ − [ξ, η] for some curve η in g that vanishes at the endpoints.
If the reduced Legendre transformation ξ 7→ µ = δlδξ is invertible, then the
Euler-Poincare´ equations are equivalent to the (right) Lie-Poisson equations
[51]:
µ˙ = −ad∗δh
δµ
µ,
where the reduced Hamiltonian is given by h(µ) = (µ, ξ)− l(ξ). These equa-
tions are equivalent to Hamiltonian equations on T ∗G relative to the Hamil-
tonian H : T ∗G 7→ R, obtained by right translating h from the identity
element to other points via the right action of G on T ∗G.
5
CHAPTER 1. INTRODUCTION 1.3. PARTICLE SOLUTIONS
1.3 Particle solutions
The Euler fluid equation has a striking feature: it admits the particle solu-
tions. The following explanation is quite basic and can be found in any good
mathematical textbook on fluid dynamics (e.g. [23]).
Taking the curl on the first equation in (1.2) gives the differential equation
on the vortex ω ≡ ∇× u
Dω
Dt − (ω · ∇)u = 0 (1.4)
where DDt = ∂∂t +u ·∇, from which the stream function ψ(x, t) can be defined
by −4ψ = ω, so we have the relation between u and ω
u(x, t) = Kω (1.5)
for some convolution operator K depending on the dimension of the space
under consideration. We have
ω(φ(x, t), t) = ∇φ(x, t) · ω(x, 0), (1.6)
where φ(x, t) is the flow map
∂φ
∂t = u(φ(x, t), t), φ(x, 0) = x. (1.7)
Now imagine the vorticity in a fluid is concentrated in N vortices
ω =
N∑
j=1
Γjδ(x− xj),
then the stream function is (here we take x = (x1, x2) ∈ R2 as an example):
ψ(x) = −
∫
ω(x′)G(x− x′)dx′ = − 12pi
N∑
j=1
Γj log ||x− xj ||, (1.8)
where G(x) = 12pi log ||x|| is the Green’s function of the Laplacian operator
4 in R2. The velocity field generated by these N point vortices is
u(x, t) = (∂x2ψ,−∂x1ψ) =
[
−
N∑
j=1
Γj
2pi
x2 − x2j
r2j
,
N∑
j=1
Γj
2pi
x1 − x1j
r2j
]
, (1.9)
6
CHAPTER 1. INTRODUCTION 1.3. PARTICLE SOLUTIONS
where rj = ||x − xj ||. Then the equation of motion that the points xj =
(x1j (t), x2j(t)), j = 1, 2, · · · , N satisfies is
dx1j
dt = −
1
2pi
∑
i6=j
Γi(x2j − x2i )
r2ij
and
dx2j
dt =
1
2pi
∑
i6=j
Γi(x1j − x1i )
r2ij
(1.10)
where rij = ||xi − xj ||. This is a Hamiltonian system with the Hamiltonian
H = − 14pi
∑
i6=j
ΓiΓj log ||xj − xi||.
Through the above process, we reduce the Euler fluid equation into a 2N ODE
system, this is quite interesting and leads directly to the particle method in
the numerical simulation we will mention in the next section. What makes it
more attractive is that not only does the Euler fluid equation have the particle
solutions as above, the other Euler equations also have this property.
The Euler fluid equation has two forms: one is (1.2), the other is (1.4)
with (1.5). Let us look at the generalised Euler equations, take the CH
equation
mt + umx + 2umx = a∂3xu with m = (1− ∂2x)u
as an example. This form of the CH equation corresponds to (1.4) with (1.5),
and if we express m in terms of u, then we get a differential equations in u,
which corresponds to the Euler fluid equation (1.2). The so-called “peakon
solutions ” in the CH equation is exactly the counterpart of “point vortices”
solutions in the Euler fluid dynamics. There is a natural parallel relation
between the Euler fluid equation and the other generalised Euler equations.
Now that the point vortices are interesting and important in the study of
fluid dynamics, we naturally want to find the role of the Dirac δ solution in
the generalised Euler equation and how the smooth solutions of the equation
tend to it? This motivates us to the asymptotic study which we do in Chapter
5.
Another thing worth mentioning: up to now, all the studies on the solitons
or the peakons are related to the complete integrability, which is a very
beautiful and wonderful but very special property that the KdV equation, the
CH equations and some other special PDEs have. But for the mCH equations
we study in this thesis, we do not know if they are completely integrable
(actually, we tend to believe they are not, supported by our numerical study),
however, they admit the Dirac δ solutions (which we call “soliton solutions”
too). Can we find a mechanism other than the integrability that generates
the solitons?
7
CHAPTER 1. INTRODUCTION 1.4. NUMERICAL APPROACHES
1.4 Numerical Approaches
Many attempts to simulating numerically the Camassa-Holm equation can
be found in the literature. We just mention a few of them.
1.4.1 Particle Method
In recent years, particle methods have become one of the most useful and
widespread tools for approximating solutions of partial differential equations
in a variety of fields. In these methods, a solution of a given equation is rep-
resented by a collection of particles, located in points qi and carrying weights
pi. Equations of evolution in time are then written to describe the dynamics
of the location of the particles and their weights. Due to the Lagrangian
nature of the method, small scales that might develop in a solution can be
easily described with a relatively small number of particles. This property
and the fact that they are mesh-free made particle methods so attractive in
a variety of problems with extremely large deformation, moving boundaries
or discontinuities, such as hydrodynamics, electrodynamics and molecular
dynamics etc.
In the numerical study of shallow water equation, Camassa, Huang and
Lee [17] proposed an algorithm corresponding to a completely integrable
particle lattice, which has some analogies with the vortex methods for the two
dimensional Euler equations. Actually, the particles they used correspond to
a solution of the following form to the CH equation:
u(x, t) =
N∑
i=1
pi(t)e−|x−qi(t)|.
They proved the convergence of the method and furthermore introduced a
fast summation algorithm to evaluate the integrals of the particle method so
that the computational cost can be reduced from O(N2) to O(N), where N
is the number of particles.
1.4.2 Box Scheme
The box scheme in numerically solving PDEs is related to Preissmann [96].
Zhao and Qin [110] and Ascher and McLachlan [6] developed it. Ascher
and McLachlan [6] compared the box schemes with other numerical methods
when applied to the KdV equation and proved that the box scheme preserves
unconditionally the dispersion relation, which accounts for the very good
8
CHAPTER 1. INTRODUCTION 1.4. NUMERICAL APPROACHES
robustness and stability. We know that the box scheme is a multi-symplectic
scheme for the KdV equation(see next subsection for this concept).
1.4.3 Multi-symplectic Methods
One of the very active fields in numerical differential equations in the recent
two decades is the structure-preserving algorithms. Among them the so-
called symplectic method is the fastest growing area. We now have some
excellent monographs on Geometric Numerical Integration, e.g., [49],[68] etc
and the review [86].
For Hamiltonian PDE, we have the so-called multi-symplectic algorithm,
which extends the symplectic algorithm for ODEs to PDEs and preserves the
symplectic structure in both the time direction and the space direction.
There are two approaches to multi-symplectic algorithms: T. J. Bridges
and S. Reich [13, 14] proposed the Hamiltonian PDEs into the forms of first
order partial differential equations
Kzt + Lzx = ∇zS(z), (1.11)
for some skew symmetric matrices K,L, and S is a smooth function, and then
introduced some numerical schemes preserving the symplecity in both x and
t directions. Wonderful reviews on this approach and its recent developments
include [15], [55],[89] and Brett Ryland’s thesis [99].
The other approach, proposed by Marsden et al., is from the variational
principle [76, 78], where they showed the existence and preservation of the
fundamental multi-symplectic structures for Hamiltonian PDEs and can be
obtained directly from the variational principle by using the multi-symplectic
geometry.
The Camassa-Holm equation is rich in geometric structures. Kouran-
baeva and Shkoller [66] studied the second order multi-symplectic field the-
ory and showed that the multi-symplectic structure can be obtained from the
variation of the action functional, which generalised the theory of [76] from
the first order field theory to the second order field theory. They applied
their abstract formulation to the Camassa-Holm equation to get a multi-
symplectic algorithm. Recently, Cohen, Owren and Raymaud [24] proposed
two different multi-symplectic formulations for the Camassa-Holm equation
and proved that the Euler box scheme preserve the multi-symplecity, and
one of their methods behaves very well in simulating the head-on collision of
solitons.
9
CHAPTER 1. INTRODUCTION 1.5. APPLICATIONS
1.5 Applications
The theory on generalised Euler equations unifies various differential equa-
tions arisen from mathematical physics, so it has naturally found many ap-
plications in the field of mathematical physics, [5, 21, 52, 54] etc. At the
same time, it has many significant applications in other fields such as image
processing [53, 85] and so on. In [85], the authors first studied the singular
solutions of the general Euler equations, and their connection with the vortex
sheet solutions of the incompressible Euler equations, and then analysed the
stability of straight and circular sheets, and studied the stability for various
metrics, and for various directions of the momentum vectors on the sheet.
D. Holm et al. [53] found some applications of EPDiff (ie the generalised
Euler equation) in the Computational Anatomy (CA) which was pioneered
by Grenander [46] through the notion of deformable templates. Roughly
speaking, a deformable template is an “object, or examplar” I0 ⊂ R2 or R3
on which a group G acts and generates a set of new objects through the orbit
GI0. The authors of [53] drew parallels between the template matching in
CA and the fluid flows in hydrodynamics. They pointed out that, the Green
kernels for the operators used to defined the quadratic cost or effort function
in CA are typically smoother than the inverse of the Helmholtz operator
1−4 used in the H1 model in hydrodynamics.
1.6 Thesis Preview
With the motivations mentioned in the beginning of this chapter, we are
concerned with the following topics in this thesis:
 the derivation of the Euler equation with respect to the general Hk(S)
metric, k ≥ 2;
 the well-posedness of the derived equation;
 some numerical approaches to this equation;
 some asymptotic analysis for the equation;
 the four particle system of the H2 metric; and
 some local well-posedness for higher dimensional case.
More specifically, Chapter 2 is concerned with the preliminary mathemat-
ics which provides us with some functional, geometrical and numerical tools
for later development.
10
CHAPTER 1. INTRODUCTION 1.6. THESIS PREVIEW
Chapter 3 is about the derivation of the generalised Euler equation on
Diff(S) with respect to the general Hk metric and its well-posedness. Section
3.1 is about the derivation of the mCH equation; Section 3.2 is about the local
well-posedness while Section 3.3 about the global well-posedness; Section 3.4
is concerned with the global weak solution of the mCH. Then we discuss
some generalisations in Sections 3.5 and 3.6. In Section 3.7, we discuss the
existence of the conjugate points of the geodesic curve starting from constant
solutions of the mCH equation.
Chapter 4 presents the numerical study on the equations. Here we con-
sider the limiting case, ie, the case of a = 0 just for simplifying the presenta-
tion. We propose two schemes: particle method (for the circle case) and the
box scheme (for the whole real line R1 case). The first corresponds to the
important properties of mCH equations: they admit the particle solutions
and the Hamiltonian structure. The latter scheme is known to be multi-
symplectic when applied to the KdV equation and some other Hamiltonian
PDEs, but we don’t know whether it is so when applied to mCH. However,
it gives a very stable simulation.
Chapter 5 studies the blowup profile by asymptotic analysis. In section
5.1, we derive some asymptotic PDE by the method of asymptotic expansion,
and then discuss the stability of the stationary solutions. Then in Section
5.3, we show some numerical simulations that suggest the blowup profile may
wander within the one parameter family of steady solutions, which we can
not explain why.
We use the particle method to study a four particle system corresponding
the mCH in Chapter 6. We know that the mCH has the following conserved
quantities:
∫
m,
∫
mu (and the third one
∫
|m| 12 for the limiting case). We
want to know if there are some other conserved quantities? From our study of
the four particle system, it is expected that there is another conserved quan-
tity because the numerical simulation on the Lyapunov exponents strongly
suggests that at most one of the Lyapunov exponents is positive which means
there should be another conserved quantity!
Higher dimensional case is studied in Chapter 7, where we have ob-
tained some local existence for the two dimensional case by the regularisation
method.
Chapter 8 outlines some future work on this equation.
At last, there are two appendices, we prove some very nice analytic prop-
erties for the Green functions for the operator 1−∂2x+∂4x in the first appendix;
and then some material on the multi-symplectic formulation for mCH, which
is obtained via the multi-symplectic geometry approach.
11
CHAPTER 1. INTRODUCTION 1.6. THESIS PREVIEW
Let us conclude this chapter with a table which describes the similari-
ties and differences between the Camassa-Holm equation and the modified
Camassa-Holm equations.
12
C
H
A
PT
ER
1.
IN
T
RO
D
U
C
T
IO
N
1.6.
T
H
E
SIS
P
R
E
V
IE
W
The Camassa-Holm equation vs the modified Camassa-Holm equation for k ≥ 2
The Camassa-Holm equation The Modified Camassa-Holm equation
Completely integrable X ? but we tend to believe it is not
X
Local well-posedness X (Kato theory) Kato theory but an extra effort needed to
get the nice properties of the Green functions
Finite time blowup X for some initial values never
Global well-posedness X for + or − initial values always X (the proof is surprisingly simple)
Number of infinite conserved quantities two obvious conserved, and our study
conserved quantities because of the integrability suggests another conserved integral
Particle methods X X
Box scheme X and multi-symplectic X
but we don’t know if it is multi-symplectic
Conjugate Points X X
to constant solutions but we need to solve a higher order linear PDE
Asymptotic profile only numerical simulation numerical results and partial analysis results
13
Chapter 2
Preliminary Tools
fµ“óõÙ¯§7k|Ùì......”
——5Ø6§úc480c"
The mechanic, who wishes to do his work well, must first sharpen
his tools...
—Confucius, The Confucian Analects, ∼ 480 BC.
A long time ago, when younger and rasher mathematicians, we both
momentarily harboured the ambition that one day, older and wiser,
we might write a multivolume treatise titled “On the Mathematical
Foundations of Numerical Analysis”. Then it dawned that such a
creation already exists: it is called “a mathematics library”.
—B. Baxter and A. Iserles [8], 2003.
In this chapter, we collect some basic concepts and theorems from par-
tial differential equations, Lie groups and Riemannian geometry, mainly for
getting acquaintance with the basic notations, ideas and terminology.
2.1 PDE Basics
We will present some fundamental concepts in PDE and Kato theory on
quasilinear evolutionary equations. The material can be found, e.g. in [37],
[61] and [102] etc.
14
CHAPTER 2. PRELIMINARY TOOLS 2.1. PDE BASICS
2.1.1 Sobolev Spaces
Definitions
Fix 1 ≤ p ≤ ∞, k a non-negative integer and Ω ⊂ Rn a domain.
Definition 2.1 The Sobolev space W k,p(Ω) consists of all locally summable
functions u : Ω 7→ R such that for each (non-negative integer-components)
multi-index α = (α1, α2, · · · , αn) with |α| ≡
∑n
i=1 αi ≤ k, the weak deriva-
tives Dαu exists and Dαu ∈ Lp(Ω). When p = 2, we usually write Hk(Ω) =
W k,2(Ω).
With the standard norms
||u||W k,p(Ω) ≡




∑
|α|≤k
∫
Ω
|Dαu|pdx


1
p
(1 ≤ p ≤ ∞)
∑
|α|≤k
esssup|Dαu| (p =∞),
(2.1)
it is easy to check that W k,p(Ω) are Banach spaces and Hk(Ω) are Hilbert
spaces if we identify f = g as an element of W k,p(Ω) provided f(x) = g(x)
for almost all x ∈ Ω.
When Ω = Rn or Tn(= Rn/Zn) and p = 2, we have another equivalent
approach to Sobolev spaces:
u ∈ Hk(Rn) or Hk(Tn)⇐⇒ both u and 〈ξ〉kuˆ ∈ L2,
where 〈ξ〉 ≡ (1 + |ξ|2) 12 and uˆ is the Fourier transform of u if Ω = Rn and uˆ
is the corresponding Fourier coefficients of u if Ω = Tn. This approach can
be easily extended from integer k to general real s ∈ R :
Hs(Rn) ≡
{
u ∈ S ′(Rn) : both u and 〈ξ〉suˆ ∈ L2(Rn)
}
,
where the Schwarz space S ′ stands for the tempered growth distributions in
Rn.
We can introduce the Sobolev spaces with negative index by the dual
space, ie, for non-negative real s ≥ 0, we define
W−s,p(Rn) ≡
(
W s,p′(Rn)
)∗
,
where 1p + 1p′ = 1. For p = 2, we have
H−s(Rn) ≡ (Hs(Rn))∗.
15
CHAPTER 2. PRELIMINARY TOOLS 2.1. PDE BASICS
Basic Properties of Sobolev Spaces
Theorem 2.2 Let Ω be bounded, and suppose that u ∈ W k,p(Ω) for some
1 ≤ p <∞, then there exist functions um ∈ C∞(Ω)
⋂
W k,p(Ω) such that
lim
m→∞
||um − u||W k,p(Ω) = 0.
This is Theorem 2 in Section 5.3.2 of [37]. We denote W k,p0 (Ω) the closure
in W k,p(Ω) of the set C∞0 (Ω) which consists of the smooth functions with
compact support in Ω.
If Ω = S is the unit circle, then we can define the Sobolev space W k,p(S)
as the completion of the smooth function space C∞(S) with respect to the
corresponding norm given by (2.1).
For Sobolev spaces, the most important theorem is the following embed-
ding theorem:
Theorem 2.3 (Embedding Theorem)
(1) If s > n2 + k for an integer k ≥ 0, then Hs(Rn) ⊂ Ck(Rn).
(2) If s > n2 + α for a real 0 ≤ α ≤ 1, then Hs(Rn) ⊂ Cα(Rn).
(3) If Ω is a bounded domain with locally Lipschitz boundary, then W k,p(Ω)
can be compactly embedded into Cλ,α(Ω) if k − np > λ + α for some
integer λ ≥ 0 and 0 ≤ α < 1. W k,p(Ω) can be compactly embedded into
Lq(Ω) if k − np > −nq .
(4) W 1,2(R1) ⊂ L∞(R1) and W 1,1(R1) ⊂ L∞(R1) for the whole real line R.
Here, the Ho¨lder spaces Cα(Ω) and Cλ,α(Ω) are defined by
Cα(Ω) ≡
{
u ∈ C(Ω) : ||u||Cα = sup
x 6=y∈Ω
|u(x)− u(y)|
|x− y|α < +∞
}
Cλ,α(Ω) ≡
{
u ∈ C(Ω) : Dλu ∈ Cα(Ω)
}
.
Proof The proof of the first three items can be found in, eg, [37], Theorem
6 in Section 5.6 of [37] and Theorem 1 in Section 5.7 there. The last item
can be proved very easily as follows: For any u ∈ W 1,2(R1), the Cauchy
inequality gives us
2u(x)2 =
∫ x
−∞
2uuxdx−
∫ +∞
x
2uuxdx ≤ ||u||2W 1,2(R1),
16
CHAPTER 2. PRELIMINARY TOOLS 2.1. PDE BASICS
so
||u||2L∞ ≤
1
2 ||u||
2
W 1,2(R1).
If u ∈ W 1,1(R1), then
|u(x)| =
∣∣∣∣
∫ x
−∞
uxdx
∣∣∣∣ ≤
∫ x
−∞
|ux|dx,
so
||u||L∞ ≤ ||u||W 1,1(R1).
From the embedding theorem, we know that if u ∈ Hs(Rn) for s > n2 ,
then u is bounded and continuous, whose dual proposition tells us that
The Dirac delta function δ ∈ H−n2−ε(Rn), for all ε > 0. (2.2)
Properties of Hs(S)
We are mainly concerned with, in the later chapters, the Sobolev spaces
Hs(S) which consists of the periodic one-variable Sobolev functions. We
need some estimates on the product of two Sobolev functions and on the
commutators which we collect here without detail proof.
Lemma 2.4 [62] For the 1D periodic Sobolev functions on S = [0, 2pi]:
(1) Hs(S) is a Banach algebra for s > 1/2, and
||uv||Hs ≤ ||u||Hs||v||Hs for u, v ∈ Hs. (2.3)
(2) For any s > 32 and u ∈ Hs(S) we have
〈u, uux〉Hs ≤ C(s)||ux||L∞||u||2Hs, (2.4)
here 〈·, ·〉s means the standard inner product in Hs(S).
If we denote Λ2k ≡ (1 − ∂2x + · · · + (−1)k∂2kx )
1
2k for integer k ≥ 1, then
Λ2k2k : Hr(S) 7→ Hr−2k(S) is an invertible mapping, and ∂xΛ2k = Λ2k∂x. We
denote Λ2k by Λ if no confusion occurs. Moreover, if we denote [Λ, f ]g =
Λ(fg)− fΛg, then from [62] [103], we have
Lemma 2.5 If s > 0 and 1 < p <∞,
(1) W s,p(S)⋂L∞(S) is an algebra, and
||uv||W s,p ≤ C(||u||L∞||v||W s,p + ||u||W s,p||v||L∞). (2.5)
17
CHAPTER 2. PRELIMINARY TOOLS 2.1. PDE BASICS
(2)
||[Λs, u]v||Lp ≤ C(p, s)(||ux||L∞||Λs−1v||Lp + ||Λsu||Lp||v||L∞). (2.6)
(3)
||[Dα, u]v||L2 ≤ C(||u||Hk||v||L∞ + ||ux||L∞||v||Hk−1) (2.7)
for all α ≤ k with k positive integer.
Kato and Ponce [62] proved this lemma for k = 1 and the same method
yields the result for the general case.
Lemma 2.6 On the norm estimates of product of two functions on S, we
have
(1) [61] Let s, t be real numbers such that −s < t ≤ s, then
C||f ||Hs||g||Ht ≥



||fg||Ht if s > 1/2,
||fg||Hs+t−1/2 if s < 1/2,
(2.8)
where C is a positive constant depending on s, t.
(2) [62] For s ≥ 0, we have
||fg||Hs ≤ C (||f ||L∞||g||Hs + ||f ||Hs||g||L∞) . (2.9)
(3) For s ≥ 0, we have
||fg||Hs ≤ C(||f ||W s,∞||g||L2 + ||g||Hs||f ||L∞). (2.10)
Proof of (3) The inequality (2.9) is Lemma X4 from [62] whose proof
is based on a lemma due to R. Coifman and Y. Meyer. Actually, the same
ideas with obvious modifications yield a proof of (2.10).
Another formal approach suggested by T. Tao [101] (page 338) is that we
can heuristically think
〈Λ〉s(fg) v g(〈Λ〉sf) + f(〈Λ〉sg), (2.11)
then take L2 norm on both sides and use the Ho¨lder inequality to get the
required inequality (2.10).
Lemma 2.7 For any two functions f, g defined on S, we have
18
CHAPTER 2. PRELIMINARY TOOLS 2.1. PDE BASICS
(1) ||fg||Ht ≤ C||f ||Ht||g||Ht for t > 12 ;
(2) ||fg||Ht ≤ C||f ||L∞||g||Ht for t ≤ 0;
(3) ||fg||Ht ≤ C||f ||Ht+1/2||g||Ht+1/2 for 0 < t ≤ 12 .
(4) ||fg||Ht ≤ C(||f ||L∞||g||Ht + ||g||L∞||f ||Ht) for t > 0.
Proof (1) is the consequence of the fact that H t is a Banach algebra for
t > 12 .
(2) For t ≤ 0, and any h ∈ H−t(S), we have
∣∣∣∣
∫
S
fghdx
∣∣∣∣ ≤ |f |L∞
∫
|gh| dx
≤ |f |L∞||g||Ht||h||H−t,
(2.12)
from which (2) follows.
(3) The inequality follows from (1) and the fact ||fg||Ht ≤ ||fg||Ht+1/2.
(4) The inequality is the Lemma X4 in [62], see also Lemma 2.6.
The similar proof of Lemmas A.2 and A.3 in [61] will give
Lemma 2.8 If s > 1/2 + 1, then
||[Λs4, f ]Λ1−s4 || ≤ C||f ′||Hs−1, (2.13)
where || · || on the left denotes the operator norm in L2(S).
Lemma 2.9 Let u ∈ Hs for some s > 3/2. Then
||Λ−r1[Λr1+r2+1, u]Λ−r2||B(L2) ≤ C||u′||Hs−1, |r1|, |r2| ≤ s− 1. (2.14)
Kato [61] proved this lemma for Λ = (1−∂2x)
1
2 , but a obvious modification
yields the same results for the general Λ = (1− ∂2x + · · ·+ (−1)k∂2kx )
1
2k and
we omit the details.
2.1.2 Kato Theory
Our local well-posedness is based on the Kato Theory [61] which sets up the
abstract theorems on the quasilinear evolutionary equations



∂u
∂t + A(t, u)u = f(t, u) ∈ X, t ≥ 0,
u(0) = u0 ∈ Y,
(2.15)
19
CHAPTER 2. PRELIMINARY TOOLS 2.1. PDE BASICS
where A(t, u) is a linear operator depending on the unknown u and X, Y
are two Banach spaces satisfying some conditions. The equations we are
concerned with in this thesis can be put into the form of
(C)



∂u
∂t + A(u)u = f(u) ∈ X, t ≥ 0,
u(0) = u0 ∈ Y.
(2.16)
That means, the operators A and f do not depend on t explicitly.
Some notations: B(X, Y ) denotes the space of all bounded linear op-
erators from a Banach space X to a Banach space Y (B(X) if X = Y );
∂ = ∂x = ∂∂x ; Λs2 = (I − ∂2x)s/2,Λs4 = (I − ∂2x + ∂4x)s/4, s ∈ R; 〈 , 〉s for
the inner product on Hs; H∞ =
⋂
s≥0Hs; [A,B] = AB − BA denotes the
commutator of the linear operators A and B.
Some assumptions on (C):
(X) X and Y are Hilbert spaces where Y ⊂ X is dense and the inclusion
continuous, and there is an isomorphism S from Y to X such that
||ω||Y = ||Sω||X for all ω ∈ Y.
(A1) Let W ⊂ Y be an open ball centered at 0. The linear operator A(u)
belongs to G(X, 1, β) where β is a real number, where G(X, 1, β) is
defined as the set of linear operators A(u) satisfying:
1. 〈Aω, ω〉X ≥ −β||ω||2X, ∀ ω ∈ D(A), the domain of A.
2. (A+ λ) is onto for some (all) λ > β.
(A2) The map
ω ∈ W 7→ B(ω) = [S,A(ω)]S−1 = SAS−1 −A ∈ B(X) (2.17)
is uniformly bounded and Lipschitz continuous, i.e., there exist con-
stants λ1, µ1 > 0 such that
||B(ω)||B(X) ≤ λ1, ||B(ω)− B(ν)||B(X) ≤ µ1||ω − ν||Y
for all ω, ν ∈W.
(A3) Y ⊆
⋂
ω∈W
D(A(ω)), so that A(ω)|Y ∈ B(Y,X) by the Closed Graph
Theorem. Moreover, there exists µ2 > 0 such that, for all ω, ν ∈ W ,
we have
||A(ω)− A(ν)||B(Y,X) ≤ µ2||ω − ν||X .
20
CHAPTER 2. PRELIMINARY TOOLS 2.2. RIEMANNIAN GEOMETRY
(f1) f : W → Y is bounded and there exist a constant µ3 > 0 such that
||f(ω)− f(ν)||X ≤ µ3||ω − ν||X , ∀ ω, ν ∈W,
||f(ω)− f(ν)||Y ≤ µ3||ω − ν||Y , ∀ ω, ν ∈W,
Theorem 2.10 (Kato [61]) Under assumptions above on (C) with u0 ∈W ,
there exists a T > 0 such that there exits a unique solution u ∈ C([0, T ], Y )∩
C1([0, T ], X) to (C). Moreover, the map u0 ∈ Y → u ∈ C([0, T ], Y ) is
continuous in the following sense: suppose
lim
n→∞
||An(ω)− A∞(ω)||B(Y,X) = 0, limn→∞ ||Bn(ω)− B∞(ω)||B(X) = 0,
lim
n→∞
||fn(ω)− f∞(ω)||Y = 0, limn→∞ ||u0,n(ω)− u0,∞(ω)||Y = 0,
and consider the Cauchy problems
(Cn)



∂un
∂t + An(un)un = fn(un) ∈ X, t ≥ 0,
un(0) = u0,n ∈W, n ∈ Z ∪ {∞}.
(2.18)
Suppose the assumptions above hold also for (Cn) with the same X, Y, S,W
and the constants β, λi, µi are independent of n. Let Tn be the time of ex-
istence of un. Then all un, with n large enough, can be extended to [0, T∞]
and
lim
n→∞
||un(t)− u∞(t)||C([0,T∞];Y ) = 0.
The proof of Theorem 2.1 can be found in [61].
2.2 Riemannian Geometry
We will state some basic concepts in Riemannian geometry which we will use
in the subsequent chapters.
Definition 2.11 Let M be a differentiable manifold modelled on Rm, and if
there is a positive definite mapping
g : TM × TM 7→ R1
which is called the Riemannian metric, then (M, g) is called a Riemannian
manifold.
21
CHAPTER 2. PRELIMINARY TOOLS 2.2. RIEMANNIAN GEOMETRY
The fundamental theorem of Riemannian geometry states that on any
Riemannian manifold there is a unique torsion-free metric connection, called
the Levi-Civita connection of the given metric. Here a metric (or Rieman-
nian) connection is a connection which preserves the metric tensor.
More precisely:
Let (M, g) be a Riemannian manifold (or pseudo-Riemannian manifold),
then there is a unique connection ∇ which satisfies the following conditions:
(1) for any vector fields X, Y, Z on M , we have,
∂Xg(Y, Z) = g(∇XY, Z) + g(Y,∇XZ),
where ∂Xg(Y, Z) denotes the derivative of the function g(Y, Z) along
the vector field X.
(2) For any vector fields X, Y on M , we have
∇XY −∇YX = [X, Y ]
where [X, Y ] denotes the Lie brackets for vector fields X, Y.
(The first condition expresses the fact that the connection is compatible with
the Riemannian metric, so that the metric tensor is preserved by parallel
transport, while the second condition expresses the fact that the torsion of
the connection is zero.)
Definition 2.12 A parametrised curve γ : I 7→ M is a geodesic at t0 ∈ I
if Ddt(
dγ(t)
dt ) ≡ ∇dγ(t)dt
dγ(t)
dt = 0 at the point t0; if γ is a geodesic at all t ∈ I, we
say γ is a geodesic curve.
Definition 2.13 The curvature R of a Riemannian manifold M is a corre-
spondence that associates to every pair of vector fields X, Y ∈ TM a mapping
R(X, Y ) : TM 7→ TM given by
R(X, Y )Z = ∇Y∇XZ −∇X∇Y Z +∇[X,Y ]Z, Z ∈ TM,
where ∇ is the Levi-Civita connnection.
Now we can introduce Jacobi fields along a geodesic γ which, in some sense,
describe the spreading rate of the nearby geodesics close to γ.
Definition 2.14 Let γ(t) be a geodesic curve on M , and if a vector field
J(t) on M satisfies the Jacobi equation
D2
dt2J(t) +R(γ
′(t), J(t))γ′(t) = 0,
then J(t) is called a Jacobi field along γ(t).
22
CHAPTER 2. PRELIMINARY TOOLS 2.3. LIE GROUPS
Clearly, γ′(t), tγ′(t) are Jacobi fields along γ(t) because R(γ′(t), γ′(t))γ′(t) =
0. So, one normally considers only the Jacobi fields along γ that are normal
to γ′.
Definition 2.15 Let γ : [0, a] 7→ M be a geodesic. The point γ(t0), with
some t0 ∈ (0, a], is said to be conjugate to γ(0) along γ if there exists a
Jacobi field J(t) along γ, not identically zero, with J(0) = J(t0) = 0.
For example, on the two dimensional sphere S2, any semicircle jointing the
north and south poles is a geodesics, and the two antipoles are conjugate
points each other. Intuitively, the existence of the conjugate point means
that the nearby geodesic curves γ1, generated by perturbating a geodesics γ
along J(t), will comes back to γ(t) at some time.
We conclude this section with the introduction of Lie derivative.
Definition 2.16 Let X be a smooth vector field on M , and T a differentiable
tensor field of rank (p, q) on M , then the Lie derivative of T with respect to
X is defined at the point p ∈M by
(£XT )p =
d
dt
∣∣∣∣
t=0
(φt)∗Tφt(p)
where φt (or φ) is the one parameter flow generated by X and φ∗ is the
pull-back of φ.
For example, £Xf = Xf is the directional derivative of f along the direc-
tional field X for any smooth function f : M 7→ R1. One can easily check
that, £XY = [X, Y ] for vector fields X, Y on M. We have also the Cartan’s
Magic Formula:
£Xω = dıXω + ıX(dω)
for any form ω on M , where d is the exterior differential operator and ıXω
the contraction of ω and X.
2.3 Lie Groups
Lie groups play a fundamentally important role in the study of differential
equations with some (continuous) symmetry [92]. We pick up some basic
concepts and results on Lie groups in this section, mainly from [5].
23
CHAPTER 2. PRELIMINARY TOOLS 2.3. LIE GROUPS
2.3.1 Lie Group and Adjoint Representation
Definition 2.17 A group G is called a Lie group if G has a smooth structure
and the two group operations are smooth. Its tangent space g at the identity
e, as a linear space, is called the vector space of the Lie algebra of G.
For example, all rotations of a rigid body about the origin is the Lie group
SO(3).
We know a coordinate change C in R4 leads to the similar transform
which sends a matrix B ∈ SO(3) to the matrix CBC−1 ∈ SO(3). A similar
structure exists for a general Lie group G.
Definition 2.18 Given a Lie group G and g ∈ G, the map Ag : G 7→ G
defined by
Ag : h 7→ ghg−1
is called an inner automorphism of G.
Obviously, Ag(e) = geg−1 = e, so the derivative of the mapping Ag maps g
to g.
Definition 2.19 The derivative Adg of Ag at the identity e:
Adg : g 7→ g, Adga = (Ag∗|e)a, a ∈ g = TeG
is called the group adjoint operator of G. Here F∗|x : TxM 7→ TF (x)M is the
derivative (or push-forward) of a mapping F : M 7→M at x.
If we denote GL(g) the space consisting of the invertible linear operators
from g to g, then the mapping Ad : g 7→ Adg ∈ GL(g) can be thought as a
mapping from G 7→ GL(g). So we can introduce
Definition 2.20 Suppose g(t) is a curve in a Lie group G, g(0) = e, g˙(0) =
ξ ∈ g, then the differential ad of the mapping Ad at the identity e
ad = Ad∗|e : g 7→ GL(g), adξ =
d
dt
∣∣∣∣
t=0
Adg(t)
is called the adjoint representation of the Lie algebra.
For example, for the special orthogonal group G = SO(n) in Rn, we have
adξω = [ξ, ω] = ξω − ωξ
is the commutator of skew symmetric matrices ξ, ω ∈ so(n).
24
CHAPTER 2. PRELIMINARY TOOLS 2.3. LIE GROUPS
Definition 2.21 The commutator in the Lie algebra g is defined as the op-
eration [ , ] : g × g 7→ g that associates to a pair of vectors ξ, η ∈ g the
vector adξη, ie,
[ξ, η] = adξη.
The tangent space g with the operation [ , ] is called the Lie algebra of the
Lie group G.
Figure 2.3.1: The vector ξ in the Lie algebra g
It is not difficult to verify that the operation [ , ] is bilinear, skew sym-
metric, and satisfies the Jacobi identity, ie, for any ξ, η, ω ∈ g, we have:
[λξ + νω, η] = λ[ξ, η] + ν[ω, η] λ, ν ∈ R or C;
[ξ, η] = −[η, ξ];
[[ξ, η], ω] + [[η, ω], ξ] + [[ω, ξ], η] = 0.
Definition 2.22 For a vector ξ ∈ g, the set Orbit(ξ) = {Adgξ : g ∈ G} is
called the adjoint (group) orbit of ξ.
One can easily find that the vectors advu, v ∈ g form the tangent space
to the adjoint orbit of the point u ∈ g.
For example, the adjoint orbit of a matrix, regarded as an element of
the Lie algebra of all complex matrices, is the set of matrices with the same
Jordan normal form.
25
CHAPTER 2. PRELIMINARY TOOLS 2.3. LIE GROUPS
2.3.2 Co-adjoint Representation of a Lie Group
When working with the Eulerian hydrodynamics, we are not dealing with
the Lie algebra g and its adjoint representation, but with its dual space g∗
and the co-adjoint representation.
Definition 2.23 (1) Denote G, g, g∗ as above, then for every g ∈ G, ξ ∈
g∗, ω ∈ g, we can define a mapping
Ad∗g : g∗ 7→ g∗
by
(Ad∗gξ)(ω) = ξ(Adgω).
The corresponding map Ad∗ : G 7→ GL(g∗) is called the co-adjoint
representation of the Lie group G.
(2) For ξ ∈ g∗, the set Orbit(ξ) ≡ {Ad∗gξ : g ∈ G} is called the co-adjoint
orbit of ξ.
(3) The co-adjoint representation of v ∈ g is the operator ad∗v : g∗ 7→ g∗
defined by
ad∗vω(u) = ω(advu) = ω([v, u])
for u, v ∈ g, ω ∈ g∗.
Clearly, the co-adjoint representation ad∗vω for v ∈ g, form the tangent space
to the co-adjoint orbit of the point ω.
2.3.3 Invariant Metrics of Lie Groups
A Riemannian metric 〈·, ·〉 on a Lie group G is left-(right-)invariant if it is
conserved under any left (or right) translation Lf (or Rf ), which means that
for any ξ, η ∈ g and g ∈ G, we have
〈Lg∗ξ, Lg∗η〉g = 〈ξ, η〉.
So we can uniquely define the left-(right-)invariant metric on the whole group
G by translating from the Lie algebra g, ie, once we know a quadratic form
on g, we can define a invariant metric on G.
Let A : g 7→ g∗ be a symmetric definite operator that defines the inner
product
〈ξ, η〉 = (Aξ, η) = (Aη, ξ)
26
CHAPTER 2. PRELIMINARY TOOLS 2.3. LIE GROUPS
for any ξ, η ∈ g. Here the round brackets stand for the standard pairing of
the elements of g and g∗. The operator A is called the inertia operator. For
any g ∈ G, A induces an operator Ag : TgG 7→ T ∗gG by
Agξ = L∗−1g ALg−1∗ξ, for ξ ∈ TgG.
We have the very illustrative commutative diagram Figure 2.3.2 from [5]:
Any tangent vector ξ ∈ TgG can be translated by Rg−1 or Lg−1 to the Lie
algebra g and we obtain two different vectors in g.
Definition 2.24 For ξ ∈ TgG, we call the two vectors
ωc = Lg−1∗ξ ∈ g, ωs = Rg−1∗ξ ∈ g
the angular velocity and spatial angular velocity respectively, here the sub-
script c in ωc stands for “corps”=body. Clearly, ωs = Adgωc.
Similarly, for the vector m = Agξ ∈ T ∗gG, we have two different vectors in
g∗:
mc = L∗gm ∈ g∗, ms = R∗gm ∈ g∗
which are called angular momentum relative to the body and the angular
momentum relative to the space. We note that mc = Ad∗gms. It is easy to
check that, for a motion g(t) ∈ G, the energy E ≡ 12〈g˙, g˙〉 can be expressed
as
E = 12〈g˙, g˙〉 =
1
2〈ωc, ωc〉 =
1
2(Aωc, ωc) =
1
2(mc, ωc) =
1
2(m, g˙).
Euler proved for G = SO(3) that any geodesic curve g(t) ∈ G must satisfy
dms
dt = 0, or equivalently
dmc
dt = ad
∗
ωcmc.
Arnold [3, 5] pointed out that the proofs are almost literally extendable to
the general case. That is, we have the abstract Euler Theorem (or general
Euler equations):
Theorem 2.25 Let G be a Lie group with a left-invariant metric, and g∗ its
Lie algebra, then any geodesic curve g(t) in G must satisfy
dms
dt = 0, or equivalently
dmc
dt = ad
∗
ωcmc. (2.19)
For a Lie group with a right-invariant metric, we have
Theorem 2.26 Let G be a Lie group with a right-invariant metric, and g∗
its Lie algebra, then any geodesic curve g(t) in G must satisfy
dms
dt = 0, or equivalently
dmc
dt = −ad
∗
ωcmc. (2.20)
27
CHAPTER 2. PRELIMINARY TOOLS 2.3. LIE GROUPS
g g
TgG
T ∗gG
g∗ g∗
Adg
A
Lg−1∗ Rg−1∗
Ag
L∗g
R∗g
Ad∗g
ωc ωs
g˙
m
mc ms
Adg
A
Lg−1∗ Rg−1∗
Ag
L∗g
R∗g
Ad∗g
Figure 2.3.2: Diagram of the operators in g and g∗
28
CHAPTER 2. PRELIMINARY TOOLS 2.3. LIE GROUPS
2.3.4 Applications to Hydrodynamics
LetM be a domain in Rn, and Diff(M) denotes the set of C∞-diffeomorphisms
on M , and Diffvol(M) for the volume-preserving C∞-diffeomorphisms on M .
Then both Diff(M) and Diffvol(M) are Lie groups if we take the composition
of two diffeomorphisms as the group multiplication operation. Diff(M) and
Diffvol(M) are the configuration manifolds of compressible and incompress-
ible fluid flows respectively.
Remark If one is concerned with the Cr-diffeomorphisms on M , then
neither Diffrvol(M) nor Diffr(M) is a Lie group although they are both con-
tinuous groups. The reason is that the left translation Lf defined by Lf(g) ≡
f ◦ g is only continuous but not smooth. The detailed explanations can be
found, eg, in [34].
29
Chapter 3
Well-posedness
Bfµ“<k^^§#Ã^^"”
——5<m­6§úc300c"
All men know the utility of useful things; but they do not know the
utility of futility.
—Zhuang Zi, ∼ 300 BC.
In this chapter, we will first derive the generalised Euler equation with
respect to the Hk, k ≥ 2 metric on the Bott-Virasoro group D̂(S) :
mt + 2uxm+ umx + a∂3xu = 0 on S, with m = Λ2k2ku, (3.1)
where the operator Λs2k = (1− ∂2x + · · ·+ (−1)k∂2kx )
s
2k , k is a positive integer
and a ∈ R (here we use this cumbersome notation just to indicate that Λs2k is
a s-th order pseudo-differential operator), then study its well-posedness and
other analytical properties of its solution.
3.1 Derivation of the Equations
3.1.1 Bott-Virasoro Group
LetDs(S) be the group of orientation preserving Sobolev Hs diffeomorphisms
of the unit circle S, and let Vects(S) = TeDs(S) be the corresponding Lie
algebra. We assume s to be large enough so that all our formal calculations
can be rigorously justified.
The Bott-Virasoro group D̂s(S) is the non-trivial central extension of
Ds(S) which is defined as follows: the set
D̂s(S) ≡ Ds(S)×R1
30
CHAPTER 3. WELL-POSEDNESS 3.1. DERIVATION OF THE EQUATIONS
and the group operation is defined by Bott [12]
ηˆ ◦ ξˆ =
(
η ◦ ξ, α+ β +
∫
S
log ∂x(η ◦ ξ)d log ∂xξ
)
, (3.2)
where ηˆ = (η, α), ξˆ = (ξ, β) with η, ξ ∈ Ds(S) and α, β ∈ R.
The corresponding Virasoro algebra V̂ect(S) is the tangent space of D̂s(S)
at the identity which is the non trivial extension of Vect(S), the tangent space
of Ds(S) at the identity of Ds(S). The commutator in the Lie algebra is given
by [5]
[V̂ , Ŵ ] ≡ −
(
(v∂xw − w∂xv)
∂
∂x, c(v, w)
)
, (3.3)
where c(v, w) ≡
∫
S
v∂3xwdx, V̂ =
(
v ∂∂x, a
)
, Ŵ =
(
w ∂∂x, b
)
with a, b ∈ R
and v ∂∂x , w ∂∂x ∈ TeDs(S).
3.1.2 Derivation of the Equations
In order to derive the equation (3.1), according to Theorem 2.26, the key
point we need to find is the formula of ad∗.
Let Û = (u ∂∂x , a), V̂ = (v ∂∂x , b), Ŵ = (w ∂∂x , c) ∈ V̂ect
s(S), and define the
Hk inner product on V̂ects(S) by
(Û , V̂ )Hk =
∫
S
(uv + uxvx + · · ·+ ∂kxu∂kxv)dx+ ab, (3.4)
then we find ad∗Û by direct calculations
(ad∗Û V̂ , Ŵ )Hk = (V̂ , adÛŴ )Hk = (V̂ , [Û , Ŵ ])Hk
= (v, uxw − uwx)Hk − b · c(u, w)
= (Λ2k2kv, uxw − uwx)L2 − b · c(u, w)
= (g + b∂3xu, w)L2 = (Λ−2k2k (g + b∂3xu), w)Hk,
(3.5)
where g = 2uxΛ2k2kv + uΛ2k2kvx. So
ad∗Û V̂ =
(
Λ−2k2k (2uxΛ2k2kv + uΛ2k2kvx + b∂3xu)
∂
∂x, 0
)
. (3.6)
31
CHAPTER 3. WELL-POSEDNESS 3.2. LOCAL WELL-POSEDNESS
The group D̂s(S) is a right-invariant group, so by Theorem 2.26, the gener-
alized Euler equation ddt Û = −ad
∗
Û Û on the Virasoro group gives us
dΛ2k2ku
dt = −(2uxΛ
2k
2ku+ uΛ2k2kux + a∂3xu),
da
dt = 0, (3.7)
which is (3.1) for m = Λ2k2ku.
We can put the equation (3.7) in the Hamiltonian form:
mt = −(m∂x + ∂xm + a∂3x)
δH
δm, with H =
1
2
∫
umdx. (3.8)
The KdV equation and the CH equation can be also put into this form (3.8)
but with m = u and m = (1 − ∂2x)u respectively. On the other hand, we
know that the KdV equation can be expressed as
mt = −∂x
δH1
δm , with m = u, H1 =
1
2
∫ (1
3u
3 − au2x
)
dx, (3.9)
and the Camassa-Holm equation
mt = −∂x(1− ∂2x)
δH1
δm , with H1 =
1
2
∫ (
u(u2 + u2x)− au2x
)
dx. (3.10)
These equations give the second Hamiltonian structure for the KdV and
CH equations respectively, where the term “bi-Hamiltonian structure” in
some literature comes from, and the bi-Hamiltonian structure leads to the
integrability of the equations and gives infinite conserved quantities for KdV
and CH. Another interesting point is that they yield a constant Poisson
structure K = ∂x, ∂x(1− ∂xx) for KdV and CH, which we do not know if it
is true also for the general k > 1 case.
3.2 Local Well-posedness
We will establish the well-posedness of (3.1) for k = 2 and the similar result
holds for the general case k ≥ 2.
Theorem 3.1 Let k = 2, u0 ∈ Hs(S), s > 2k − 12 = 7/2. Then, there exist
a T > 0 depending on ||u0||s, and a unique solution u satisfying (3.1) in the
distribution sense such that
u ∈ C([0, T ], Hs(S)) ∩ C1([0, T ], Hs−1(S)).
Moreover, the map u0 ∈ Hs 7→ u ∈ C([0, T ], Hs(S)) is continuous.
32
CHAPTER 3. WELL-POSEDNESS 3.2. LOCAL WELL-POSEDNESS
We can rewrite (3.1) for k = 2 in two ways:



mt = −umx − 2mux − a∂3xu, x ∈ S, t ∈ R,
m(x, 0) = m0(x) = Λ44u0(x),
(3.11)
where m = Λ44u = (I − ∂2x + ∂4x)u, Λs4 = (I − ∂2x + ∂4x)
s
4 . Or



ut = −uux − ∂xΛ−44 (u2 + 12u2x − 72u2xx − 3ux∂3xu)− a∂3xΛ
−4
4 u, x ∈ S, t ∈ R,
u(x, 0) = u0(x).
(3.12)
If we denote A(u) = u∂x, f ≡ −∂xΛ−44 (u2 + 12u2x − 72u2xx − 3ux∂3xu) −
a∂3xΛ−44 u, then (3.12) has the form of (2.16):
(C)



∂u
∂t + A(u)u = f(u) ∈ X, t ≥ 0,
u(0) = u0 ∈ Y.
(3.13)
In order to use the Kato’s theory on quasilinear evolutionary equations to
prove this theorem, we need to verify that the conditions in Kato’s Theorem
are satisfied, ie, we need the following lemmas:
Lemma 3.2 The operator A(u) = u∂x, with u ∈ Hs, s > 32 belongs to
G(Hs−1, 1, β) for some β > 0.
Proof According to the definition of G(Hs−1, 1, β) in the assumption
(A1) in Section 2.1.2 we need to verify two conditions:
(1)
〈A(u)φ, φ〉Hs−1 ≥ −β||φ||2Hs−1, (3.14)
(2) A(u) + λ is onto Hs−1 for for some λ > β.
We use Λ for Λ4 in this proof just for simplifying the notations.
〈u∂xφ, φ〉Hs−1 = 〈Λs−1(u∂xφ),Λs−1φ〉L2 for φ ∈ Hs−1(S)
= 〈[Λs−1, u]∂xφ+ u∂xΛs−1φ,Λs−1φ〉L2
= 〈[Λs−1, u]∂xφ,Λs−1φ〉L2 − 12〈ux, (Λs−1φ)2〉L2
(3.15)
33
CHAPTER 3. WELL-POSEDNESS 3.2. LOCAL WELL-POSEDNESS
The first term can be estimated as follows:
〈[Λs−1, u]∂xφ,Λs−1φ〉L2
≤ ||Λs−1, u]∂xφ||L2||Λs−1φ||L2
≤ C(||ux||L∞||Λs−2∂xφ||L2 + ||Λs−1u||L2||∂xφ||L∞)||Λs−1φ||L2
(by Lemma 2.5)
≤ C||u||Hs||Λs−1φ||2L2.
(3.16)
It is easier to get the estimate for the second term:
〈ux, (Λs−1φ)2〉L2 ≤ C||ux||L∞||Λs−1φ||2L2
≤ C||u||Hs||Λs−1φ||2L2.
(3.17)
Now we are going to verify the second condition: A(u) + λ is onto Hs−1 for
some λ > β. Clearly, A(u) + λ is a closed operator, so we need only to prove
that A(u) + λ has dense range for λ > β.
Suppose ψ ∈ Hs−1(S) such that
〈(A(u)+λ)φ, ψ〉Hs−1 = 0 for any φ ∈ D(A(u)) = {φ ∈ Hs−1 : u∂xφ ∈ Hs−1},
(3.18)
then we have in the distributional sense
−∂x(uΛ2s−2ψ) + λΛ2s−2ψ = 0,
which yields
〈Λ2s−2ψ, uψx〉(H1−s,Hs−1) + λ〈Λ2s−2ψ, ψ〉(H1−s,Hs−1) = 0, (3.19)
here (H1−s, Hs−1) stands for the natural pairing between H1−s = (Hs−1(S))∗
and Hs−1 and Λ2s−2 : Hs−1(S) 7→ H1−s(S) is an isomorphism. On the other
hand, by the inequality (3.14), we have
〈Λ2s−2ψ, uψx〉(H1−s,Hs−1) = 〈ψ, uψx〉Hs−1 ≥ −β||ψ||2Hs−1, (3.20)
so the above two equations and the condition λ > β imply that ψ ≡ 0. This
means, A(u) + λ has dense range for λ > β.
34
CHAPTER 3. WELL-POSEDNESS 3.2. LOCAL WELL-POSEDNESS
Lemma 3.3 B(u) = [Λ14, u∂x]Λ−14 ∈ B(Hs−1) for u ∈ Hs, s > 3/2.
Proof Obviously, [Λ, u∂x]Λ−1w = [Λ, u]Λ−1wx, so we have
||B(u)w||Hs−1 = ||Λs−1[Λ, u]Λ−1wx||
= ||Λs−1[Λ, u]Λ1−sΛ(s−2)wx||
≤ C||ux||Hs−1 ||Λ(s−2)wx||L2
≤ C||u||Hs||w||Hs−1.
(3.21)
In the second-to-last estimate, we have used Lemma 2.9.
From the proof, we have
||(B(u)− B(v))w||Hs−1 ≤ C||w||Hs−1||u− v||Hs for u, v ∈ Hs. (3.22)
Lemma 3.4 For u ∈ Hs(S) with s > 3/2,
(a) Hs ⊂ D(u∂x) = {f ∈ Hs−1 : u∂xf ∈ Hs−1}, s > 3/2.
(b) u∂x ∈ B(Hs, Hs−1), s > 3/2.
(c) ||u∂x − v∂x||B(Hs,Hs−1) ≤ C||u− v||s−1.
Proof For f ∈ Hs−1 we have
||u∂xf ||Hs−1 ≤ C||u||Hs−1||∂xf ||Hs−1 because Hs−1 is a Banach algebra
≤ C||u||Hs−1||f ||Hs
(3.23)
which proves the first two parts. The third part follows directly from the
above inequality.
Lemma 3.5 Let f(u) = −∂xΛ−44 (u2 + 12u2x− 72u2xx−3ux∂3xu)−a∂3xΛ
−4
4 u, s >
7/2, then for any u, v with ||u||Hs, ||v||Hs ≤ C, we have
(a) ||f(u)− f(v)||Hs−1 ≤ C||u− v||Hs−1.
(b) ||f(u)− f(v)||Hs ≤ C||u− v||Hs.
35
CHAPTER 3. WELL-POSEDNESS 3.2. LOCAL WELL-POSEDNESS
Proof In the proof, we use || · ||s to stand for || · ||Hs.
(a) We need only to verify that
||∂xΛ−44 (
7
2u
2
xx + 3ux∂3xu−
7
2v
2
xx − 3vx∂3xu)||s−1 ≤ C||u− v||s−1,
for the corresponding inequality for the other three terms is easy to verify.
||∂xΛ−44 (u2xx − v2xx)||s−1 ≤ C||∂2x(u+ v)∂2x(u− v)||s−4
≤ max{||∂2x(u+ v)||s−4, ||∂2x(u+ v)||L∞, ||∂2x(u+ v)||s−7/2}
·max{||∂2x(u− v)||s−4, ||∂2x(u− v)||s−7/2} (by Lemma 2.7)
≤ C||u− v||s−3/2 ≤ C||u− v||s−1.
||∂xΛ−44 (ux∂3xu− vx∂3xv)||s−1 ≤ C||ux∂3xu− vx∂3xv||s−4 (3.24)
= C||ux∂3xu− ux∂3xv + ux∂3xv − vx∂3xv||s−4 (3.25)
≤ C||ux(∂3xu− ∂3xv)||s−4 + ||(ux − vx)∂3xv||s−4. (3.26)
We estimate these two terms separately. If s− 4 > 12 or −12 < s− 4 ≤ 0, we
can easily get from Lemma 2.7 in Chapter 2 that
||ux(∂3xu− ∂3xv)||s−4 ≤ C||∂3x(u− v)||s−4 ≤ C||u− v||s−1. (3.27)
If 0 < s− 4 ≤ 12 , we have to use Lemma 2.6 in Chapter 2, to get
||ux∂3x(u− v)||Hs−4 ≤ C(||ux||L∞||∂3x(u− v)||Hs−4 + ||ux||W s−4,∞||∂3x(u− v)||L2)
≤ C(||u||W 1,∞||u− v||Hs−1 + ||u||W s−3,∞||u− v||H3)
≤ C||u||Hs||u− v||Hs−1 because s− 4 > 0.
(3.28)
Similarly, we can estimate the other term in (3.26). Here we just write out
the formula for the case 0 < s− 4 ≤ 12 .
||(ux − vx)∂3xv||Hs−4 ≤ C(||ux − vx||s−4||∂3xv||L∞ + ||∂3xv||s−4||ux − vx||L∞
≤ C||v||Hs||u− v||Hs−1.
(3.29)
Adding up all the above estimates yields
||f(u)− f(v)||Hs−1 ≤ C||u− v||Hs−1. (3.30)
36
CHAPTER 3. WELL-POSEDNESS 3.2. LOCAL WELL-POSEDNESS
(b) Similar situation as in (a).
||∂xΛ−44 (u2xx − v2xx)||s ≤ C||∂2x(u+ v)∂2x(u− v)||s−3
≤ C||∂2x(u+ v)||s−3||∂2x(u− v)||s−3 ≤ C||u+ v||s−1||u− v||s−1
≤ C||u− v||s,
||∂xΛ−44 (ux∂3xu− vx∂3xv)||s
≤ C||ux∂3xu− vx∂3xv||s−3 = C||ux∂3xu− ux∂3xv + ux∂3xv − vx∂3xv||s−3
≤ C||u||s−2||u− v||s + C||v||s||ux − vx||s−3
≤ C||u− v||s,
here we have used the fact that Hs is a Banach algebra for s > 1/2.
Proof of Theorem 3.1 Now Theorem 3.1 is just a direct consequence
of Kato’s Theorem with Y = Hs(S), X = Hs−1(S) and the above Lemmas.
Theorem 3.6 If the Theorem 3.1 yields the maximal time interval of exis-
tence is [0, T ), then we have T = +∞ or
lim
t→T−
||u(t)||Hs = +∞ if T <∞. (3.31)
Proof From Theorem 3.1, we have T = +∞ or
lim
t→T−
(||u(t)||Hs + ||ut(t)||Hs−1) = +∞ if T <∞. (3.32)
On the other hand, we have from the proof of Theorem 3.1 and the equation
(3.12) that
||ut(t)||Hs−1 ≤ C||u(t)||Hs−1||u(t)||Hs ≤ C||u(t)||2Hs, (3.33)
which yields what we want.
3.2.1 Conservation Laws
Based on the local well-posedness, some conservation laws can be established.
In this subsection, we assume that the solutions are smooth enough that all
the calculations can be done rigorously.
Theorem 3.7 Let u(x, t) be the solution to (3.1) with u0 ∈ H∞, and m0 =
(1 − ∂2x + ∂4x)u0,, then in the time interval of existence of u, we have the
following conserved quantities:
I1 =
∫
m =
∫
u, (3.34)
I2 =
∫
um =
∫
(u2 + u2x + u2xx). (3.35)
37
CHAPTER 3. WELL-POSEDNESS 3.3. GLOBAL WELL-POSEDNESS
Proof Integrating directly the equation (3.1), we have the first conserved
quantity. We can exploit the equations (3.11)(3.12) to verify that
dI2
dt =
∫
utmdx+
∫
umtdx = 0.
Geometrically, the fact I2 is conserved just means that velocity vector of the
geodesic curve has a constant length along the geodesics.
3.3 Global Well-posedness
The Camassa-Holm equation (1.3) can reach a singularity in a finite time if
m0 = (I−∂2x)u0 changes sign. However, this can not happen for the modified
equation (3.1) by our next theorem.
Theorem 3.8 Suppose k ≥ 2 in (3.1). If the initial value m(0, x) ∈ L2(S),
then m(t, x) ∈ L2(S) for any finite time t > 0, and there exists a constant
C0 depending only on the norm of initial values u such that
||m||L2 ≤ eC0t||m0||L2. (3.36)
Proof of Theorem 3.8 We prove the Theorem 3.8 for sufficiently
smooth function m and the general case m0 ∈ L2 follows by a standard
density argument. Multiply (3.1) by m and integrate over S, we have
1
2
d
dt ||m||
2
L2 + 2
∫
uxm2 +
∫
ummx = a
∫
m∂3xu, (3.37)
Clearly,
∫
m∂3xu =
∫
∂3xuΛ2k2ku = 0. So
d
dt ||m||
2
L2 = −3
∫
m2ux, (3.38)
from which
d
dt ||m||
2
L2 ≤ 3|ux|L∞||m||2L2. (3.39)
On the other hand, I2 =
∫
S umdx is a conserved quantity for (3.1), ie
k∑
l=0
||∂lxu(t, x)||2L2 =
k∑
l=0
||∂lxu(0, x)||2L2. (3.40)
38
CHAPTER 3. WELL-POSEDNESS 3.3. GLOBAL WELL-POSEDNESS
So from the Sobolev embedding theorem and k ≥ 2 we have
|ux|L∞ ≤ C||uxx||L2 ≤ C0, (3.41)
where C0 is a constant depending only on the initial condition. The Gronwall
inequality and (3.39) yield
||m||L2 ≤ eC0t||m0||L2. (3.42)
For the limiting mCH, ie, a = 0, we have even better results:
Theorem 3.9 If m0 ∈ Lp(S), where 2 ≤ p < +∞, then there exists a
unique global solution m(t, x) ∈ Lp(S) to (3.1) with a = 0 and k = 2 such
that m ∈ Lp(S) and
||m(·, t)||Lp(S) ≤ eCt||m0||Lp(S), (3.43)
where C is a constant independent of p. Moreover, if m0 ∈ W 1,p(S), then
m(·, t) ∈W 1,p(S) and there exist constants C1, C2 independent of p such that
||m(·, t)||W 1,p(S) ≤ eC1e
C2t||m0||W 1,p(S) (3.44)
Proof of Theorem 3.9: We prove the conclusions for the smooth
enough data and the general case can be reached by standard approximations.
Multiplying (3.11) by |m|p−1sgnm, we have
1
p
d|m|p
dt +
1
p (|m|
pu)x + (2−
1
p)|m|
pux = 0, (3.45)
integrating this equation, we have
1
p
d||m||pLp
dt = −(2−
1
p)
∫
|m|pux
≤ (2− 1p)||ux||L∞||m||
p
Lp.
(3.46)
Combining with the embedding ||ux||L∞ ≤ C ′||uxx||L2 ≤ C, we can easily
see that
||m(·, t)||Lp ≤ eCt||m0||Lp. (3.47)
Taking the derivative with respect to x in (3.11), we have got
dmx
dt + 3mxux + 2muxx +mxxu = 0. (3.48)
39
CHAPTER 3. WELL-POSEDNESS 3.3. GLOBAL WELL-POSEDNESS
Multiplying this equation by |mx|p−1sgnmx,
1
p
d|mx|p
dt +
1
p(|mx|
pu)x + (3−
1
p)|mx|
pux + 2muxx|mx|p−1sgnmx = 0. (3.49)
Now integrating over S gives
1
p
d||mx||pLp
dt + (3−
1
p)
∫
|mx|pux + 2
∫
muxx|mx|p−1sgnmx = 0. (3.50)
So, if m0 ∈ W 1,p(S), p ≥ 1, then m0 ∈ L∞(S), and (3.47) yields that, for
some constants C1, C2 > 0,
||m(·, t)||L∞(S) ≤ C1eC2t. (3.51)
At the same time
||ux||L∞ ≤ C.
Moreover, the embedding theorems and (3.47) tell us that
||uxx||L∞ ≤ C||∂3xu||L2 ≤ C||∂4xu||L2 ≤ C(||m||L2+||u−uxx||L2) ≤ eCt. (3.52)
Now it is easy to derive from (3.50)∼(3.52) that there exist constants C1, C2
independent of p such that
||mx||Lp ≤ eC1e
C2t||m0x||Lp. (3.53)
3.3.1 Extra Properties for a = 0
Lemma 3.10 Let u(x, t) be the solution to (3.1) with a = 0, u0 ∈ H∞, and
suppose that m0 = (1− ∂2x + ∂4x)u0 ≥ 0 (or ≤ 0), then m = (1− ∂2x + ∂4x)u ≥
0 (respectively ≤ 0), moreover, if m ≥ 0, then
∫
S
m1/2dx =
∫
S
m1/20 dx.
Proof The proof of Lemma 3.3 in [28] applies here with little change, but
we include it here for completeness.
Let ε > 0,Ω ⊂ R2 be a bounded domain, v ∈ H1(Ω). Then from [44], we
know that √ε+ v+,
√ε+ v− ∈ H1(Ω), with
∇√ε+ v+ =
∇v
2√ε+ v+
χ[v > 0], ∇√ε+ v− =
∇v
2√ε+ v−
χ[v < 0],
40
CHAPTER 3. WELL-POSEDNESS 3.3. GLOBAL WELL-POSEDNESS
where χ stands for the characteristic function. Now if we take Ω = [0, 2pi]×
[0, t], then
d
dt
∫
S
√ε+m+ =
1
2
∫
S
mt√ε+m+
χ[m > 0]
= −
∫
S
mux√ε+m+
χ[m > 0]− 12
∫
S
mxu√ε+m+
χ[m > 0]
= −
∫
S
√ε+m+uxχ[m > 0] + ε
∫
S
ux√ε+m+
χ[m > 0]
−12
∫
S
mxu√ε+m+
χ[m > 0].
(3.54)
Integrating the first integral by parts ,
d
dt
∫
S
√ε+m+ = ε
∫
S
ux√ε+m+
χ[m > 0] +R(t, ε), (3.55)
where R(t, ε) =
∑
xs∈A σ(xs)
√εu(xs) with A ≡ {x ∈ S : m(x) = 0} and
σ(xs) = 1 or − 1 depending on xs is the left or right end point of the
composing intervals of {x ∈ S : m(x) > 0}. Whatever the value of R(t, ε)
is, we always have
|R(t, ε)| ≤
√
ε
∫
S
|ux|, (3.56)
and so
∣∣∣∣
d
dt
∫
S
√ε+m+
∣∣∣∣ =
∣∣∣∣ε
∫
S
ux√ε+m+
χ[m > 0] +R(t, ε)
∣∣∣∣
≤ 2
√
ε
∫
S
|ux| ≤
√
ε
(
1 +
∫
S
(u2 + u2x + u2xx)
)
.
(3.57)
So from the obvious fact that
∫
S(u2 + u2x + u2xx) is conserved, we have
∣∣∣∣
∫
S
√ε+m+ −
∫
S
√ε+m0+
∣∣∣∣ ≤
√
ε
(
1 +
∫
(u20 + u20x + u20xx)
)
t. (3.58)
Letting ε→ 0, we get ∫
S
√m+ =
∫
S
√m0+. (3.59)
Similarly we have the conservation for
∫
S
√m−, from which the lemma fol-
lows.
Remark
41
CHAPTER 3. WELL-POSEDNESS 3.3. GLOBAL WELL-POSEDNESS
(a) From the proof, we can find that the essential part in the proof is the
equation mt = −2umx− uxm and the conservation of
∫
(u2 + u2x + u2xx)
(which is
∫
um). The exact relation between m and u does not really
matter as long as
∫
|ux| can be controlled by
∫
um.
(b) If m0(x) changes sign on S, we have
∫
S
|m| 12 dx =
∫
S
|m0|
1
2 dx. (3.60)
In fact, the limiting mCH has a very nice property: the zero points of m
evolve along the characteristics, this can easily be seen from mt+umx =
0 at the points where m = 0. So on each subinterval of S where m does
not change sign, the integral
∫
|m| 12 dx over the subinterval is conserved.
There is another point of view: formally the limiting mCH can be put
in the form of
d
dt |m|
1
2 + (|m| 12u)x = 0. (3.61)
This means the so-called Casimir functional is conserved:∫
|m| 12 dx =
∫
|m0|
1
2 dx. (3.62)
Lemma 3.11 Let u0 ∈ Hs(S), s > 7/2 and m0 = (1−∂2x +∂4x)u0 ≥ 0(or ≤
0), then ∃K > 0 such that ||uxxx||L∞ ≤ K.
Proof At first, we assume that u0 ∈ H∞, u solves (3.1), then it is easy
to show that ||u||2L2 + ||ux||2L2 + ||uxx||2L2 is conserved as long as u exists as a
solution to (3.1). From Lemma 3.10, we have m = Λ44u ≥ 0 (or ≤ 0). Let
x0 ∈ S satisfy uxxx(x0) = 0, then ∀y ∈ S, we have
uxxx(y) =
∫ y
x0
∂4xudx =
∫ y
x0
(u− ∂2xu+ ∂4xu)dx−
∫ y
x0
(u− ∂2xu)dx
≤
∫
S
mdx+ ||u||L1 + ||uxx||L1 =
∫
S
m0dx+ ||u||L1 + ||uxx||L1
≤
∫
S
m0dx+ C||u||L2 + C||uxx||L2 ≤ K,
(3.63)
where K depends on m0 and ||u0||H2 and C is a constant independent of u.
Similarly, we have (here we use x0 and x0 + 2pi to stand for the same point
on the circle S assuming the circumference of S is 1 with x0 ≤ y ≤ x0 + 2pi)
−uxxx(y) =
∫ x0+2pi
y
∂4xudx ≤ K.
42
CHAPTER 3. WELL-POSEDNESS 3.3. GLOBAL WELL-POSEDNESS
So far we have proved the Lemma for u0 ∈ H∞. A standard approximation
can give the proof for u0 ∈ Hs(S), s > 7/2.
Theorem 3.12 Suppose k = 2, u0 ∈ Hs(S), s > 7/2, then Equation (3.1)
with a = 0 admits a unique solution in C([0,+∞), Hs(S))
⋂
C1([0,+∞), Hs−1(S))
if the initial momentum m0 ≥ 0.
This theorem holds also valid for k ≥ 2 as long as u0 ∈ Hs(S) with s >
2k − 12 . In order to prove Theorem 3.12, we need the following lemma:
Lemma 3.13 Assume the conditions in Theorem 3.12 hold, then ||u(t)||Hs
is finite for any 0 < t <∞.
Proof Apply Λs4 to ut = −uux − f(u), where f(u) = ∂xΛ−44 (u2 + 12u2x −7
2u2xx − 3ux∂3xu), and multiply by Λs4u and then integrate over S, we get
d
dx ||u||
2
s = −2〈u, uux〉s − 2〈u, f(u)〉s (3.64)
By Lemma 2.5, we have
|〈u, uux〉s| ≤ Cs||ux||L∞||u||2s. (3.65)
The Cauchy inequality gives
|〈u, f(u)〉s| ≤ ||u||s||f(u)||s, (3.66)
and
||f(u)||s ≤ C||u2 + 12u2x − 72u2xx − 3ux∂3xu||Hs−3
≤ C(||u2||s−3 + ||u2x||s−3 + ||u2xx||s−3 + ||ux∂3xu||s−3 + ||u||Hs)
≤ C(||u||L∞||u||s−3 + ||ux||L∞||ux||s−3 + ||uxx||L∞||uxx||s−3
+||ux||L∞||∂3xu||s−3 + ||∂3xu||L∞||ux||s−3 + ||u||Hs)
≤ C||u||s,
(3.67)
where we used Lemma 2.5 and Lemma 3.11. So we have
d
dt ||u||
2
s ≤ C||u||2s, (3.68)
and so the Gronwall’s inequality completes the proof of Lemma.
Proof of Theorem 3.12 Theorem 3.12 is a direct consequence of The-
orem 3.6 and Lemma 3.13 above.
43
CHAPTER 3. WELL-POSEDNESS 3.4. WEAK SOLUTIONS FOR A = 0
3.4 Weak Solutions for a = 0
What we have obtained is the well-posednss for u in Hs(S) with s > 2k− 12 ,
this excludes the δ-momentum solutions. But we know that the δ-momentum
solutions play a very important role in the study of Euler equations. So we
want to enlarge a little bit the solution space to include the δ-momentum
solutions.
Equation (3.12) with a = 0 can be rewritten as



ut + F (u)x = 0, x ∈ S, t ∈ R,
u(x, 0) = u0(x),
(3.69)
where
F (u) = 12u
2 + Λ−44 (u2 +
1
2u
2
x −
7
2u
2
xx − 3ux∂3xu)
= 12u
2 + Λ−44 (u2 +
1
2u
2
x −
1
2u
2
xx)− 3(∂xΛ−44 )(uxuxx).
Definition 3.14 Let u0 ∈ H2(S). A function u : [0,+∞)× S → R is called
a global weak solution to (3.69) if u ∈ C([0,∞);H2) and ∀ T > 0, we have
∫ T
0
∫
S
(uϕt + F (u)ϕx)dxdt +
∫
S
u0(x)ϕ(0, x)dx = 0, ∀ ϕ ∈ C1,c([0, T )× S),
(3.70)
where C1,c([0, T )×S) is the set of all first order smooth function with compact
support in [0, T )× S.
Theorem 3.15 Let u0 ∈ H2(S), and m0 = (I − ∂2x + ∂4x)u0 is a positive
Radon measure on S. Then there exists a unique global weak solution u ∈
C([0,∞);H2(S)) of (3.1) and such that m = Λ44u is a positive Radon measure
on S whose total variation on S is uniformly bounded for t ≥ 0. Moreover
we have
∫
S
udx =
∫
S
u0dx,
∫
S
(u2 +u2x+u2xx)dx =
∫
S
(u20 +u20x+u20xx)dx, (3.71)
Proof of Theorem 3.15 Let θ ≡ ||m0||M = ||u0 − ∂2xu0 + ∂4xu0||M be
the variation of the Radon measure m0, then by Lemma 5.2 in [28], there
exist positive functions mn0 ∈ C∞(S) such that ||mn0 ||L1 ≤ C for a constant
44
CHAPTER 3. WELL-POSEDNESS 3.4. WEAK SOLUTIONS FOR A = 0
C independent of n, and mn0 → m0 in D′(S). If we denote un0 = Λ−44 mn0 ,
then mn0 = un0 − ∂2xun0 + ∂4xun0 ,
un0 → u0 in H2(S),
and
||un0 ||2H2 =
∫
S
|un0 |2 + |un0x|2 + |un0xx|2dx = |
∫
S
mn0 · un0dx|
≤ ||mn0 ||L1||un0 ||L∞ ≤ C||mn0 ||L1 ||un0 ||H1 ,
(3.72)
which implies that
||un0 ||2H2 =
∫
S
|un0 |2 + |un0x|2 + |un0xx|2dx ≤ C||mn0 ||2L1 ≤ Cθ2. (3.73)
Then by applying Theorems 3.1 and 3.8 with the smooth initial value un0(x),
there exists a unique solution to (3.69) un ∈ C([0,∞);Hs)∩C1([0,∞);Hs−1).
From Lemma 3.10, we know that mn(t, x) > 0 if we denote mn = un−∂2xun+
∂4xun, so
||un(t)||H2 = ||un0 ||H2 ≤ C and ||mn(t)||L1 = ||mn0(t)||L1 ≤ C,
where C is a constant independent of n. Hence
||∂4xun||L1 ≤ ||un||L1 + ||∂2xun||L1 + ||mn(t)||L1 ≤ C
and
||∂3xun||L∞ ≤ C, with C independent of n.
So {un(t)} is a compact set in H2(S) for any t ≥ 0.
On the other hand, ||dundt ||H2 = ||F (un)x||H2 can be estimated as follows:
||[(un)2]x||H2 = 2||ununx||H2 ≤ C(||ununx||L2 + ||unxxunx||L2 + ||ununxxx||L2)
≤ C||∂3xun||L2 ≤ C||∂3xun||L∞ ≤ C,
(3.74)
||∂xΛ−44 (v2 + 12v2x − 72v2xx − 3vx∂3xv)||H2
≤ C||v2 + 12v2x − 72v2xx − 3vx∂3xv||H−1
≤ C if v = un.
(3.75)
45
CHAPTER 3. WELL-POSEDNESS 3.4. WEAK SOLUTIONS FOR A = 0
So ||dundt ||H2 = ||F (un)x||H2 ≤ C with C independent of t and n. So the
family un(t), as functions from [0,+∞) to H2(S), is equi-continuous. There-
fore Arzela`-Ascoli theorem ([98] or see the “ Arzela`-Ascoli Theorem” in the
Wikipedia) tells us that {un(t)}n≥1 ⊂ C([0, T ];H2) is a compact subset for
any T > 0. So we can extract a subsequence unk and there is a function
u ∈ C([0,∞);H2) such that
unk → u in C([0,∞);H2),
with
||u(t)− u(s)||H2 ≤ C|t− s|, ∀ t, s ≥ 0.
From lim
n→∞
un(t) = u(t) in H2(S) for any t ≥ 0, we have
u(0) = u0.
Taking nk →∞ in
∫ T
0
∫
S
(unkϕt+F (unk)ϕx)dxdt+
∫
S
unk0 (x)ϕ(0, x)dx = 0, ∀ ϕ ∈ C1,c([0, T )×S)
(3.76)
yields that u ∈ C([0,∞);H2) is the weak solution to (3.69).
From the proof above, we can easily get the conserved quantities and that
the total variation ||m(t, ·)||M of the limit measure m satisfies
||m(t, ·)||M ≤ ||mn(t)||L1 = ||mn0 (t)||L1 ≤ C.
Uniqueness: Now we are proving the uniqueness of the solution. Here we
just sketch the proof, and a rigorous argument can be realised by a standard
mollification method. Let G(x) be the Green’s function for the operator
Λ44 = I − ∂2x + ∂4x acting on H∞(S), then from
(I − ∂2x + ∂4x)G(x) = δ(x) =
∞∑
n=−∞
enix, (3.77)
we have
G(x) =
∞∑
n=−∞
1
1 + n2 + n4 e
inx
= 1 + 2
∞∑
n=1
1
1 + n2 + n4 cos(nx) x ∈ S.
(3.78)
Obviously, for any 0 ≤ ε < 1, G(x) ∈ C2+ε(S). Moreover, from the Appendix
A of this thesis, we can even have that ∂3xG ∈ L∞(S).
46
CHAPTER 3. WELL-POSEDNESS 3.4. WEAK SOLUTIONS FOR A = 0
Suppose u, v ∈ C([0,∞);H2) are two solutions of (3.12), ie, they both
solve the equation



ut = −uux − ∂xΛ−44 (u2 + 12u2x − 72u2xx − 3ux∂3xu), x ∈ S, t ∈ R,
u(x, 0) = u0(x).
(3.79)
Or equivalently,



ut = −uux −Gx ∗ (u2 + 12u2x − 72u2xx − 3ux∂3xu), x ∈ S, t ∈ R,
u(x, 0) = u0(x).
(3.80)
here ∗ stands for the convolution. Denote
M ≡ sup
t≥0
{||Λ44u||M + ||Λ44v||M} <∞, (3.81)
then for all t, x ∈ R+ × S, we have
||u(x, t)||L∞ = ||G ∗m||L∞ ≤ ||G||L∞||m||M ≤ CM,
||ux(x, t)||L∞ = ||Gx ∗m||L∞ ≤ CM,
||uxx(x, t)||L∞ = ||Gxx ∗m||L∞ ≤ CM,
||uxxx(x, t)||L∞ = ||Gxxx ∗m||L∞ ≤ CM,
(3.82)
and same estimates hold true for v as well.
Let w = u− v and A(u) = u2 + 12u2x − 72u2xx − 3ux∂3xu, then



wt = −uwx − wvx −Gx ∗ (A(u)− A(v)), x ∈ S, t > 0,
w|t=0 = 0. x ∈ S
(3.83)
so
d
dt
∫
S
|w|dx =
∫
S
wtsgnw
=
∫
−uwxsgnw − wvxsgnw −Gx ∗ (A(u)− A(v))sgnw.
(3.84)
47
CHAPTER 3. WELL-POSEDNESS 3.4. WEAK SOLUTIONS FOR A = 0
d
dt
∫
S
|wx|dx =
∫
S
wxtsgnwx
=
∫
−[wx(ux + vx) + uwxx + wvxx]sgnwx −Gxx ∗ (A(u)−A(v))sgnwx.
(3.85)
d
dt
∫
S
|wxx|dx =
∫
S
wxxtsgnwxx
= −
∫
[wxx(2ux + vx) + wx(2vxx + uxx) + u∂3xw + w∂3xv]sgnwxx
−
∫
Gxxx ∗ (A(u)− A(v))sgnwxx.
(3.86)
Using the estimates (3.82) for u, v, we have
∣∣∣∣
∫
−uwxsgnw − wvxsgnw
∣∣∣∣ ≤ CM
(∫
|w|+ |wx|
)
;
∣∣∣∣
∫
[wx(ux + vx) + uwxx + wvxx]sgnwx
∣∣∣∣ ≤ CM
(∫
|w|+ |wx|+ |wxx|
)
;
∣∣∣∣
∫
[wxx(2ux + vx) + wx(2vxx + uxx)]sgnwxx
∣∣∣∣ ≤ CM
(∫
|wx|+ |wxx|
)
.
(3.87)
On the other hand, A(u)− A(v) = w(u + v) + 12wx(ux + vx) −
7
2wxx(uxx +
vxx)− 3ux∂3xw − 3wx∂3xv, and integration by parts gives us
Gx ∗ (ux∂3xw) = Gxx ∗ (uxwxx)−Gx ∗ (uxxwxx);
Gxx ∗ (ux∂3xw) = Gxxx ∗ (uxwxx)−Gxx ∗ (uxxwxx)
(3.88)
which enables us to estimate
∣∣∣∣
∫
Gx ∗ (A(u)−A(v))sgnw
∣∣∣∣ ≤ CM
(∫
|w|+ |wx|+ |wxx|
)
;
∣∣∣∣
∫
Gxx ∗ (A(u)−A(v))sgnwx
∣∣∣∣ ≤ CM
(∫
|w|+ |wx|+ |wxx|
)
.
(3.89)
The other terms in (3.86) can be estimated as follows:
∫
u∂3xwsgnwxx =
∫
u ddx |wxx|dx = −
∫
|wxx|uxdx, (3.90)
48
CHAPTER 3. WELL-POSEDNESS 3.5. THE WHOLE REAL LINE CASE
so ∣∣∣∣
∫
u∂3xwsgnwxx
∣∣∣∣ ≤ CM
∫
|wxx|. (3.91)
It is easy to see
∣∣∣∣
∫
w∂3xvsgnwxx
∣∣∣∣ ≤ C||∂3xv||L∞
∫
|w| ≤ CM
∫
|w|. (3.92)
In order to estimate
∫
Gxxx ∗ (A(u) − A(v))sgnwxx, we need only estimate∫
Gxxx ∗ (ux∂3xw) because the other terms can be estimated in the same way
as the above terms. Again, the integration by parts yields
Gxxx ∗ (ux∂3xw) = Gxxxx ∗ (uxwxx)−Gxxx ∗ (uxxwxx)
= Gxx ∗ (uxwxx)−G ∗ (uxwxx) + uxwxx −Gxxx ∗ (uxxwxx)
(3.93)
here, we have used the definition of G, which gives us
Gxxxx ∗ f −Gxx ∗ f +G ∗ f = f.
Now it is clear that
∣∣∣∣
∫
Gxxx ∗ (ux∂3xw)
∣∣∣∣ ≤ CM
∫
|wxx|. (3.94)
Taking all the above estimates in account, we have
d
dt
∫
S
(|w|+ |wx|+ |wxx|)dx ≤ CM
∫
S
(|w|+ |wx|+ |wxx|)dx, (3.95)
and so the Gronwall’s inequality yields w ≡ 0. That completes the proof of
Theorem 3.15.
3.5 The Whole Real Line Case
We have discussed the well-posedness of equation (3.1) in the periodic case.
Actually, some of the above results hold true with Λ = (1−∂2x)k on the whole
real line case:
mt + 2uxm+ umx = a∂3xu in R1, with m = (1− ∂2x)ku. (3.96)
49
CHAPTER 3. WELL-POSEDNESS 3.5. THE WHOLE REAL LINE CASE
More specifically, the local well-posedness Theorem 3.1 holds true if u0 ∈
L1(R1) ∩ Hs(R1), combining our arguments here and those estimates es-
tablished for (1 − ∂2x) in [97]. Theorem 3.8 with m0 ∈ L2(R1) and u0 ∈
Hs(R1) ∩ L1(R1) holds true. Using Lemma 3.16 we are going to prove, we
can prove that Theorem 3.12 holds true for (3.96) with a = 0 if we suppose
m0 ≥ 0, u0 ∈ L1(R1) ∩ Hs(R1) with some s > 2k − 12 . If a = 0 in (3.96),
Theorem 3.15 holds true for u0 ∈ Hs(R1) ∩ L(R1) with m0 = (1 − ∂2x)2u0 a
positive Radon measure.
In fact, the only things we need to check are
(a) G(x) ≥ 0 ( x ∈ R1) for the fundamental solution G(x) of the operator
(1− ∂2x)k on R1;
(b) ||∂3xG||L∞ <∞;
(c) the proof of Lemma 3.11;
The items (a) and (b) will be proved in the Appendix A, and here we just
prove a lemma analogous to Lemma 3.11 (take k = 2 as an example).
Lemma 3.16 Let a = 0, u0 ∈ Hs(R1), s > 7/2, m0 = (1−∂2x)2u0 ≥ 0(or ≤
0) smooth enough and u0 ∈ L1(R1), then ∃K > 0 such that ||uxxx||L∞ ≤ K.
Proof From the assumption m0 = (1 − ∂2x)2u0 ≥ 0, we can prove that
m(x, t) ≥ 0 for any t ≥ 0 using the argument in Lemma 3.10, so we have
u = G ∗m ≥ 0 because G(x) > 0. From (3.96), we have
||u(t, ·)||L1(R1) = ||m(t, ·)||L1(R1) = ||m0(t, ·)||L1(R1), (3.97)
and the conservation law
∫
um =
∫
R1
(u2 + 2u2x + u2xx)dx =
∫
R1
(u20 + 2u20x + u20xx)dx, (3.98)
which implies
||ux||L∞ ≤ C (3.99)
by the Sobolev embedding theorem.
On the other hand, neither m nor u changes sign, so we have
0 ≤
∫ x
mdx =
∫ x
−∞
(u− 2∂2xu+ ∂4xu)dx ≤ ||u||L1 − 2ux + ∂3xu (3.100)
||u||L1 = ||m||L1 ≥
∫ x
−∞
(u− 2∂2xu+ ∂4xu)dx ≥ −2ux + ∂3xu, (3.101)
50
CHAPTER 3. WELL-POSEDNESS 3.6. REMARKS ON THE GENERALISATIONS
which implies
||2ux − ∂3xu||L∞ ≤ ||u||L1. (3.102)
So combining the equations (3.99)(3.102), we have
||∂3xu||L∞ ≤ C (3.103)
with a constant C depending only on the L1 norm and H2 norm of the initial
u0.
3.6 Remarks on the Generalisations
In fact, for the circle S case, the arguments in this chapter up to now can
be easily extended to the general Hk metric case (k ≥ 1) with some obvious
modifications.
Going further, from the derivation of the equation, we can find that the
specific form of the inertia operator Λ does not really matter. We can take,
for example, Λ ≡ (1 − ∂2x)k with some integer k ≥ 1 (that is what we have
done in the whole real line case), or even more generally, Λ = (1 − ∂2x)
r
2
for any real number r ≥ 0, and derive the corresponding generalised Euler
equation with respect to the Hr metric. From the arguments above, for the
limiting case a = 0, we know that as long as r > 32 , there will be no finite
time blowup phenomena.
But for the whole line R case, it seems that we need to choose Λ = (1−∂2x)k
in order to guarantee that the corresponding Green’s function is positive. We
have to impose another assumption m0 ∈ L1(R1) and some extra assumptions
on u0 as we did in the previous section.
3.7 Conjugate Points and Beyond
In this section, we turn back to the geometrical aspect of the mCH. Now
that mCH is the geodesic equation and its solution the geodesic curve, it is
natural to study the geometry of D̂(S) around the geodesic curve. Here, we
are going to exploit the sectional curvature etc to investigate the existence
of conjugate points.
Theorem 3.17 The geodesic in D̂(S) with initial conditions ηˆ(0) = (e, 0)
and ˙ˆη(0) = (v0 ∂∂x , b), where v0, b are constants, contains points conjugate to
ηˆ(0) along ηˆ.
51
CHAPTER 3. WELL-POSEDNESS 3.7. CONJUGATE POINTS AND BEYOND
Proof For the right-invariant vector fields Û =
(
u ∂∂x , a
)
, V̂ =
(
v ∂∂x , b
)
, the
covariant derivative ∇Û V̂ can be obtained from the formula ([22])
2∇Û V̂
= [Û , V̂ ]− ad∗Û V̂ − ad
∗
V̂ Û
= [Û , V̂ ]−
(
(Λ−2k2k (2uxΛ2k2kv + uΛ2k2kvx + 2vxΛ2k2ku+ vΛ2k2kux + b∂3xu+ a∂3xv) ∂∂x , 0
)
=
(
(uxv − uvx − Λ−2k2k (2uxΛ2k2kv + uΛ2k2kvx + 2vxΛ2k2ku+ vΛ2k2kux + b∂3xu+ a∂3xv) ∂∂x ,
c(u, v)) ,
(3.104)
ie
∇Û Û = −
(
(Λ−2k2k (2uxΛ2k2ku+ uΛ2k2kux + a∂3xu)
∂
∂x, 0)
)
.
On the other hand,
[[Û , V̂ ], V̂ ] =
(
((uxv − uvx)xv − (uxv − uvx)vx)
∂
∂x,−c(uxv − uvx, v)
)
,
so by the formula ([22])
R(Û , V̂ )V̂ = ∇Û∇V̂ V̂ −∇V̂∇Û V̂ −∇[Û,V̂ ]V̂ (3.105)
R(Û , V̂ ) = (R(Û , V̂ )V̂ , Û)Hk = 14 ||ad
∗
Û V̂ + ad
∗
V̂ Û ||2Hk −
(
ad∗Û Û , ad
∗
V̂ V̂
)
Hk
−34 ||[Û , V̂ ]||2Hk − 12
(
[[Û , V̂ ], V̂ ], Û
)
Hk
− 12
(
[[V̂ , Û ], Û ], V̂
)
Hk
,
(3.106)
we can get the Riemaniann curvature R(Û , V̂ )V̂ and the sectional curvature
R(Û , V̂ ) ≡ (R(Û , V̂ )V̂ , Û)Hk although the calculation is lengthy and messy.
However, if V̂ = (v0 ∂∂x , b) is a constant vector field, then the calculation is
much simpler:
∇Û V̂ = −
(
(v0Λ−2k2k ux +
1
2bΛ
−2k
2k ∂3xu)
∂
∂x, 0
)
, (3.107)
∇V̂ Û = −
(
(v0ux + v0Λ−2k2k ux +
1
2bΛ
−2k
2k ∂3xu)
∂
∂x, 0
)
, (3.108)
52
CHAPTER 3. WELL-POSEDNESS 3.7. CONJUGATE POINTS AND BEYOND
∇[Û,V̂ ]V̂ = −
(
(v20Λ−2k2k uxx +
1
2bv0Λ
−2k
2k ∂3xu)
∂
∂x, 0
)
, (3.109)
∇V̂∇Û V̂ =
(
(v20Λ−2k2k uxx +
1
2bΛ
−2k
2k ∂4xu+ v20Λ−4k2k uxx
+bvΛ−4k2k ∂4xu+
1
4Λ
−4k
2k ∂6xu)
∂
∂x, 0
)
,
(3.110)
so the Riemannian curvature
R(Û , V̂ )V̂ = ∇Û∇V̂ V̂ −∇V̂∇Û V̂ −∇[Û,V̂ ]V̂
=
(
(−14b
2Λ−4k2k ∂6xu− v20Λ−4k2k uxx − bv0Λ−4k2k ∂4xu)
∂
∂x, 0
)
.
(3.111)
and the sectional curvature
R(Û , V̂ ) = (R(Û , V̂ )V̂ , Û)Hk
= 14b
2
∫
S
∂3xuΛ−2k2k ∂3xu+ v20
∫
uxΛ−2k2k ux − bv0
∫
uxxΛ−2k2k uxx
= 14
∫
S
(
−bΛ−k2k ∂3xu+ 2v0Λ−k2k ux
)2 dx ≥ 0.
(3.112)
Let η̂(t) be the geodesic with the initial condition ˙̂η(t) = V̂ , and Ŵ (t) be
an arbitrary vector along η̂(t) and
(w(t, x) ∂∂x, s(t)) ≡ dη̂(t)Rη̂−1(t)Ŵ (t),
where Rg denote the right multiplication by g on the Virasoro group.
dη̂tRη̂−1t
(
R(Ŵ (t), ˙̂η(t)) ˙̂η(t)
)
= R
((
w(t, x) ∂∂x, s(t)
)
,
(
v0
∂
∂x, b
))(
v0
∂
∂x, b
)
= −14
((
b2Λ−4k2k ∂6xw + 4v20Λ−4k2k wxx + bv0Λ−4k2k ∂4xw
) ∂
∂x, 0
)
.
(3.113)
53
CHAPTER 3. WELL-POSEDNESS 3.7. CONJUGATE POINTS AND BEYOND
dη̂tRη̂−1t
(
∇ ˙̂ηt∇ ˙̂ηtŴ (t)
)
= ∇(v0 ∂∂x ,b)∇(v0 ∂∂x ,b)
(
w(t, x) ∂∂x, s(t)
)
= ∇(v0 ∂∂x ,b)
(
[wt − (v0wx + v0Λ−2k2k ∂xw +
1
2Λ
−2k
2k ∂3xw)]
∂
∂x, s
′(t)
)
=
(
H(t, x) ∂∂x, s
′′(t)
)
,
(3.114)
where
H(t, x) = ∂2xw − 2v0wtx + v20wxx − 2v0Λ−2k2k wxt − bΛ−2k2k ∂t∂3xw + 2v20Λ−2k2k wxx
+bv0Λ−2k2k ∂4xw + v20Λ−4k2k wxx + bv0Λ−4k2k ∂4xw +
1
4b
2Λ−4k2k ∂6xw.
(3.115)
Then the Jacobi equation along η̂(t)
∇ ˙̂η(t)∇ ˙̂η(t)Ŵ (t) +R(Ŵ (t), ˙̂η(t)) ˙̂η(t) = 0 (3.116)
reads s′′(t) = 0 and
∂2w
∂t2 − 2v0
∂2w
∂t∂x + v
2
0
∂2w
∂x2 + 2v
2
0Λ−2k2k
∂2w
∂x2 − 2v0Λ
−2k
2k
∂2w
∂t∂x
−bΛ−2k2k
∂4w
∂t∂x3 + bv0Λ
−2k
2k
∂4w
∂x4 = 0
(3.117)
that is
( ∂
∂t − v0
∂
∂x
)2
w−2v0Λ−2k2k
( ∂
∂t − v0
∂
∂x
)
wx−bΛ−2k2k
( ∂
∂t − v0
∂
∂x
)
∂3xw = 0.
(3.118)
For any integer n ≥ 1, if we denote k(n) = (1 + n2 + n4 + · · ·+ n2k)−1,
µ = nv0k(n)−
1
2bk(n)n
3 and λ = nv0 + nv0k(n)−
1
2bk(n)n
3,
then a direct calculation tells us that
w(t, x) = sin(µt) sin(nx+ λt), s(t) ≡ 0,
54
CHAPTER 3. WELL-POSEDNESS 3.8. CONCLUSIONS
is a non-trivial solution to the Jacobi equation (3.117). Clearly, Ŵ is always
perpendicular to ˙̂η(t), so it is a Jacobi field along η̂(t). If we take
t = 2pijµ for j = 0,±1,±2, · · ·
we get the points conjugate to η̂(0), which completes the proof of Theorem
3.17.
Remark From the theorem 3.17, we can obtain that the constant so-
lutions are stable. However, if we are concerned only with the stability of
the constant solutions, we can use the energy method to give a very simple
proof.
Theorem 3.18 Any constant solution (v0, b) is nonlinearly stable.
Proof Let m = m0 = v0 be the constant solution, we can introduce the
functional
H1(m) =
1
2
∫
S
umdx−
∫
S
v0mdx, (3.119)
then it is easy to check that
δH1
δm
∣∣∣∣
m0
= 0, δ
2H1
δm2
∣∣∣∣
m0
= Λ−2k2k > 0, (3.120)
which yields that m0 is a local strict minimum point of H1.
3.8 Conclusions
We have studied various analytical properties of the one dimensional mCH.
We have first derived the one dimensional mCH according to Arnold’s view-
point in Section 3.1, then have exploited the Kato theory to establish its local
well-posedness in Hs(S) with s > 2k − 12 in Section 3.2. After that, we have
proved that ||m||L2 is always finite in finite time t if the initial momentum m0
belongs to L2(S), which means that the solutions of the mCH will not blow-
up in finite time if the initial value is smooth enough. This is totally different
from the Camassa-Holm equation which may admit some finite time blow-up
solution even for some very smooth initial values. Then we have studied the
extra properties for the limiting mCH, ie, the case of a = 0. In this case, we
have proved that the mCH admits a unique solution u ∈ C([0,+∞), Hs(S))∩
C1([0,+∞), Hs−1(S)) if the initial momentum does not change sign (here
m0 may not be in L2(S)). In Section 3.4, we have introduced the notion of
55
CHAPTER 3. WELL-POSEDNESS 3.8. CONCLUSIONS
weak solution which includes the δ-momentum solutions, and then proved
its well-posedness for the limiting mCH with the initial momentum being
positive Radon measures, ie, Theorem 3.15, by an approximation process. I
guess the assumption a = 0 is only of technique significance and the similar
results hold true also for the general case a 6= 0 although I have not yet
found a proof. The difficulty here is that for the case a 6= 0, we can not
have
∫
|m| =
∫
|m0| from the conservation of
∫
m. Then I have made some
remarks on the generalisations of the previous results to the whole line R1
case under some mild extra assumptions, and to different inertia operators in
Sections 3.5, 3.6 respectively. Here in order to get the required positivity of
the Green’s function G, we have to switch to the inertia operator (1− ∂2x)k.
Then, we have looked at the geometry of D̂(S) around the geodesic curve
and proved that the geodesic in D̂(S) has conjugate points to the starting
point.
56
Chapter 4
Numerics
“A major task of mathematics is to harmonize the continuous and
the discrete, to include them in one comprehensive mathematics,
and to eliminate obscurity from both.”
—E. T. Bell, Men of Mathematics, 1965.
In this chapter, we will consider the numerical methods for the limiting
case of mCH for k = 2, ie,



mt = −umx − 2mux
m(x, 0) = m0(x).
(4.1)
Why do we consider the case a = 0? This is because in the study of the CH
equation, the limiting case is the most interesting case. Moreover, our study
on weak solutions in Chapter 3 is restricted to the limiting case. So we focus
here on the limiting case a = 0 for the mCH.
We will consider (4.1) on the circle S in Section 4.1 and on the whole real
line R1 in Section 4.2.
4.1 Particle methods
The point vortex algorithm is one of the most efficient methods in the study
of ideal hydrodynamics. Similarly, we can introduce the particle method in
the computation of mCH. The basic idea is as follows: from Theorem 3.15,
we know (4.1) has solutions supported at points on S via the following sum
57
CHAPTER 4. NUMERICS 4.1. PARTICLE METHODS
over Dirac delta measures,
m(t, x) =
N∑
i=1
pi(t)δ(x− qi(t)), (4.2)
and the velocity
u(t, x) = G ∗m =
N∑
i=1
pi(t)G(x− qi(t)) (4.3)
is the superposition of the velocity of each “soliton” supported at qi(t), where
G is the fundamental solution of the inertia operator Λ. Plugging (4.2)(4.3)
into (4.1), we have got an ODE of 2N variables:



q˙i = h
N∑
j=1
G(qi − qj)pj , i = 1, 2, · · · , N,
p˙i = −hpi
N∑
j=1
G′(qi − qj)pj , i = 1, 2, · · · , N.
(4.4)
Camassa et al. [17] proved that the particle method is convergent for the
Camassa-Holm equation, their proof used the complete integrability of the
equation although then mentioned that the use of integrability is perhaps
“overkill”. We can see that the reduction of the PDE (4.1) to the ODE (4.4)
is not related to the integrability. We will show that it applies to the modified
CH equation too. Our proof is similar to those of [17] except that we do not
use the integrability of its discretization to prove the global existence of the
corresponding ODEs.
Let G(x) be the Green’s function for the operator Λ44 = I−∂2x +∂4x acting
on H∞(S), then from
(I − ∂2x + ∂4x)G(x) = δ(x) =
∞∑
n=−∞
enix, (4.5)
we have
G(x) =
∞∑
n=−∞
1
1 + n2 + n4 e
inx = 1+2
∞∑
n=1
1
1 + n2 + n4 cos(nx) x ∈ S. (4.6)
Obviously, for any 0 ≤ ε < 1, G(x) ∈ C2+ε(S).
58
CHAPTER 4. NUMERICS 4.1. PARTICLE METHODS
Now that
u(x, t) =
∫ 2pi
0
G(x− y)m(y, t)dy (4.7)
and if m0 ≥ c > 0, then from Lemma 3.10 we have m(x, t) ≥ 0 for any t > 0,
so (4.1) can be rewritten as
(m1/2)t = −(um1/2)x . (4.8)
Let us introduce an auxiliary function
w(x, t) =
∫ x
0
m(y, t)1/2dy, (4.9)
then wxt + (uwx)x = 0, ∀ x ∈ S. So there exists a function g(t) such that
wt + uwx = g(t) ∀ x ∈ S. (4.10)
Introducing characteristic curves
x = q(ξ, t), q(ξ, 0) = ξ, (4.11)
then Equation (4.10) reads as
x˙ = q˙ = u(q, t), w˙ = g, (4.12)
where f˙ denotes the total derivative
f˙ ≡
( ∂
∂t + u
∂
∂x
)
f.
From (4.12), we have
w(q(ξ, t), t) =
∫ t
0
g(s)ds+ w(ξ, 0), and so dwdξ =
dw0
dξ , (4.13)
where w0(ξ) ≡ w(ξ, 0) and dwdξ is uniquely determined by m. Combining (4.7)
with the first equation of (4.12) gives
u(q(ξ, t), t) = q˙(ξ, t) =
∫ 2pi
0
G(q(ξ, t)− q(η, t))m(q(η, t), t)∂q(η, t)∂η dη. (4.14)
From (4.9) and (4.13) we have
m(q(ξ, t), t) =
( dw
dξ
∂q(ξ,t)
∂ξ
)2
=
( dw0
dξ
∂q(ξ,t)
∂ξ
)2
. (4.15)
59
CHAPTER 4. NUMERICS 4.1. PARTICLE METHODS
Introducing an auxiliary function
p(ξ, t) = m(q(ξ, t), t)∂q(ξ, t)∂ξ =
(w′0(ξ))2
∂q(ξ,t)
∂ξ
, (4.16)
then we have
p˙(ξ, t) = −p(ξ, t)
∫ 2pi
0
G′(q(ξ, t)− q(η, t))p(η, t)dη, (4.17)
and (4.14) becomes
q˙(ξ, t) =
∫ 2pi
0
G(q(ξ, t)− q(η, t))p(η, t)dη. (4.18)
The solution to (4.17)(4.18) with the initial conditions q(ξ, 0) = ξ, p(ξ, 0) =
(w′0(ξ))2 determines the characteristic curves x = q(ξ, t). On the other hand,
(4.17)(4.18) is a Hamiltonian system with
H = 12
∫
S×S
G(q(ξ, t)− q(η, t))p(ξ, t)p(η, t)dξdη.
Integrating directly (4.17) yields that
P ≡
∫ 2pi
0
p(ξ, t)dξ
is independent of time t because G′(x) is symmetric with respect to x = pi.
From (4.17) we have |p˙(ξ, t)/p(ξ, t)| ≤ c1P, where c1 = |G′(x)|L∞. So
p(ξ, 0)e−c1Pt ≤ p(ξ, t) ≤ p(ξ, 0)ec1Pt, ∀ξ ∈ S (4.19)
In order to approximate the Hamiltonian equations (4.17)(4.18):



q˙(ξ, t) =
∫ 2pi
0
G(q(ξ, t)− q(η, t))p(η, t)dη,
p˙(ξ, t) = −p(ξ, t)
∫ 2pi
0
G′(q(ξ, t)− q(η, t))p(η, t)dη,
(4.20)
we can use the so-called particle method, which takes
qi(t) ≡ qi(ξi, t), pi(t) ≡ pi(ξi, t), i ∈ N
60
CHAPTER 4. NUMERICS 4.1. PARTICLE METHODS
as position coordinates and momenta, and if, for example, q and p are eval-
uated at points ξi = ih, i = 1, 2, · · · , N, obtain the (finite dimensional)
discretised version of (4.20):



q˙i = h
N∑
j=1
G(qi − qj)pj, i = 1, 2, · · · , N,
p˙i = −hpi
N∑
j=1
G′(qi − qj)pj, i = 1, 2, · · · , N.
(4.21)
Compared with other classical numerical methods for PDE, one of the main
advantages of the particle method is that it preserves the Hamiltonian struc-
ture of (4.20) and so we can use the geometric integration [49] to simulate it
numerically.
Proposition 4.1 For any lp > 0, the right hand side of (4.21) is Lipschitz
continuous on (q, p) ∈ D, where D ⊂ R2N is the set of points (q, p) =
(q1, q2, · · · , qN ; p1, p2, · · · , pN) :
0 ≤ qi ≤ 2pi, i = 1, 2, · · · , N, maxi |pi| < lp <∞.
So the system of ODEs (4.21) admits a unique local solution.
Proof We demonstrate only for the first equation of (4.21) and omit
the second one for it is analogous. Let (p, q), (p˜, q˜) ∈ D ⊂ R2N , c0 =
61
CHAPTER 4. NUMERICS 4.1. PARTICLE METHODS
maxx∈S |G(x)|, c1 = maxx∈S |G′(x)|, then
∣∣∣∣∣h
N∑
j=1
G(qi − qj)pj − h
N∑
j=1
G(q˜i − q˜j)p˜j
∣∣∣∣∣
≤
∣∣∣∣∣h
N∑
j=1
G(qi − qj)pj − h
N∑
j=1
G(qi − qj)p˜j
+h
N∑
j=1
G(qi − qj)p˜j − h
N∑
j=1
G(q˜i − q˜j)p˜j
∣∣∣∣∣
≤
∣∣∣∣∣h
N∑
j=1
G(qi − qj)pj − h
N∑
j=1
G(qi − qj)p˜j
∣∣∣∣∣
+
∣∣∣∣∣h
N∑
j=1
G(qi − qj)p˜j − h
N∑
j=1
G(q˜i − q˜j)p˜j
∣∣∣∣∣
≤ c0h
N∑
j=1
|pj − p˜j|+ h
N∑
j=1
|G(qi − qj)−G(q˜i − q˜j)| |p˜j|
≤ c0h
N∑
j=1
|pj − p˜j|+ c1hmaxj |p˜j|
(
N |qi − q˜i|+
N∑
j=1
|qj − q˜j |
)
So if we denote ||v|| =
N∑
i=1
|vi| for v ∈ RN , L = max{c0h, c1hlpN}, then we
have the wanted estimate:
∣∣∣∣∣h
N∑
j=1
G(qi − qj)pj − h
N∑
j=1
G(q˜i − q˜j)p˜j
∣∣∣∣∣ ≤ L(||p− p˜||+ ||q − q˜||). (4.22)
For the global in time existence we have
Proposition 4.2 If the initial momenta are positive, pi ≥ ε > 0, i =
1, 2, · · · , N, then the solution to (4.21) exists uniquely for all times.
Proof From the Hamiltonian structure of (4.21), it is not difficult to prove
that P = h
∑N
i=1 pi is independent of time t. From the second equation of
(4.21), we have | p˙ipi | ≤ c1P, where c1 = max |G
′(x)|, so
pi(0)e−c1Pt ≤ pi(t) ≤ pi(0)ec1Pt. (4.23)
If the initial momenta are positive, then pi(t) are positive and bounded for
all times t < +∞. So the global existence follows.
62
CHAPTER 4. NUMERICS 4.1. PARTICLE METHODS
Now we will prove that the solution of (4.21) converges to that of (4.20)
as h→ 0.
Let q(ξ, t), p(ξ, t) be the solution to (4.20) with the initial data
q(ξ, 0) = ξ, p(ξ, 0) = p0(ξ), (4.24)
while q˜(t), p˜(t) stand for the solution to (4.21) with
q˜i(0) = q(ξi, 0) = ξi, p˜i(0) = p0(ξi). (4.25)
qi(t) = q(ξi, t), pi(t) = p(ξi, t) denotes the PDE solution evaluated at the grid
points, φi = qi − q˜i, ψi = pi − p˜i and ||φ|| = h
∑N
i=1 |φi|, ||ψ|| = h
∑N
i=1 |ψi|
denote the l1 norm. From (4.19)(4.23) we easily know that for any T > 0,
there exists a constant PT <∞, independent of h, such that
max{p˜i(t) : 1 ≤ i ≤ N ; 0 ≤ t ≤ T},max{p(ξ, t) : ξ ∈ S; 0 ≤ t ≤ T} ≤ PT
for h small enough (or equivalently, for N large enough).
Theorem 4.3 Consider (4.20) with (4.24) and (4.21) with (4.25). If p0(ξ) >
0, ξ ∈ S, smooth enough, then for any finite time T > 0, there exists a grid
length h such that
||φ(t)||+ 1PT
||ψ(t)|| ≤ Ch
2
PT
(eC′PT t − 1) (4.26)
for 0 ≤ t ≤ T, where C, C ′ are constants independent of T and h.
Proof Because the two equations have the same initial values at the grid
points, so
|q˜i(t)− q(ξi, t)| ≤
∫ t
0
∣∣∣∣∣h
N∑
j=1
G(qi(s)− qj(s))pj(s)
−
∫
S
G(q(ξi, s)− q(η, s))p(η, s)dη
∣∣∣∣ds
+ h
∫ t
0
∣∣∣∣∣
N∑
j=1
G(q˜i(s)− q˜j(s))p˜j(s)−
N∑
j=1
G(qi(s)− qj(s))pj(s)
∣∣∣∣∣ ds.
(4.27)
The first integral of the right hand side is controlled above by Ch2t because
the Riemannian sum h
∑N
j=1G(qi(s)−qj(s))pj(s) is the composite trapezoidal
63
CHAPTER 4. NUMERICS 4.1. PARTICLE METHODS
approximation of the integral
∫
S G(q(ξi, s)−q(η, s))p(η, s)dη. The second one
is estimated as follows (letting c0 = max |G(x)|, c1 = max |G′(x)|.):
h
∣∣∣
∑N
j=1G(q˜i(s)− q˜j(s))p˜j(s)−
∑N
j=1G(qi(s)− qj(s))pj(s)
∣∣∣
≤ h
∣∣∣
∑N
j=1G(q˜i(s)− q˜j(s))p˜j(s)−
∑N
j=1G(q˜i(s)− q˜j(s))pj(s)
∣∣∣
+h
∣∣∣
∑N
j=1G(q˜i(s)− q˜j(s))pj(s)−
∑N
j=1G(qi(s)− qj(s))pj(s)
∣∣∣
≤ c0h
∑N
j=1 |pj(s)− p˜j(s)|+ c1h
∑N
j=1 (|qi(s)− q˜i(s)|+ |qj(s)− q˜j(s)|) pj(s)
≤ c0h
∑N
j=1 |pj(s)− p˜j(s)|+ c1PTh
∑N
j=1 (|qi(s)− q˜i(s)|+ |qj(s)− q˜j(s)|) ,
so
|φi| ≤
∫ t
0
(
c0h
N∑
j=1
|ψj |+ c1PTh
N∑
j=1
(|φi|+ |φj|)
)
ds+ Ch2t, (4.28)
and hence
||φ(t)|| ≤
∫ t
0
(c0||ψ||+ 2c1PT ||φ||) ds + Ch2t. (4.29)
Similarly, we have (letting c2 = max |G′′(x)|)
|ψi(t)| ≤
∫ t
0
(2c1PTh|pi(s)− p˜i(s)|+ c2piPTh
N∑
j=1
(|qi(s)− q˜i(s)|
+|qj(s)− q˜j(s)|))ds+ Ch2t,
(4.30)
and hence
||ψ(t)|| ≤
∫ t
0
(
2c1hPT ||ψ(s)||+ 2c2PT 2||φ(s)||
)
ds+ Ch2t, (4.31)
from which we have
||φ(t)||+ 1PT
||ψ(t)|| ≤
∫ t
0
(2(c1 + c2)PT ||φ||+ (c0 + 2c1h)||ψ||) ds+ Ch2t
= 2(c1 + c2)PT
∫ t
0
(||φ||+ 1PT
||ψ||)ds
+(2c1h+ c0 − 2c1 − 2c2)
∫ t
0 ||ψ||ds+ Ch2t
≤ 2(c1 + c2)PT
∫ t
0
(||φ||+ 1PT
||ψ||)ds+ Ch2t,
(4.32)
64
CHAPTER 4. NUMERICS 4.2. BOX SCHEME
as long as h < 1 + 2c2−c02c1 ! (It’s easy to verify that 1 +
2c2−c0
2c1 is indeed a
positive number.) Now (4.32) and Gronwall inequality yield
||φ(t)||+ 1PT
||ψ(t)|| ≤ Ch
2
2(c1 + c2)PT
(e2(c1+c2)PT t − 1) (4.33)
for 0 < t < T.
Remark The convergence proof of the particle method here is similar
to that in [17]. But for the CH equation, in order to establish the global
well-posedness of the reduced particle system, Camassa et al. [17] used the
complete integrability, although they mentioned that this property may be
overkill. But for the mCH, we know that the solution will not blowup in finite
time and so the global existence of the corresponding ODE system follows
without the use of complete integrability.
Remark Similar to the vortex method in hydrodynamics [9, 10], the
particle algorithm has several distinctive features:
 the interactions of the “solitons” mimic the physical mechanisms de-
scribed by the original PDE;
 there are no inherent errors which behave like the numerical viscosity
of conventional Eulerian difference methods. Such diffusive errors can
overtake the effects of physical viscosity in high Reynolds number flow
simulation.
 One can use the geometric numerical integrators [49] to solve the re-
sulted ODE to preserve the geometric structure that the ODE has.
4.2 Box Scheme
In this section, we propose a so-called box scheme to solve the equation:



mt = −umx − 2mux on R1,
m(x, 0) = m0(x).
(4.34)
The box scheme dated back to Preissmann [96] and was mathematically
developed by Zhao and Qin [110]. It is a nondissipative scheme and exten-
sively used in the computational fluid dynamics. Ascher and McLachlan [6]
have compared the box scheme and other geometric integrators for the KdV
65
CHAPTER 4. NUMERICS 4.2. BOX SCHEME
equation and have analysed the dispersion relation to give an explanation of
the stability of the box scheme when applied to hyperbolic systems.
Introduce the finite difference operators Dx, Dt and the mean operator
M :
Dxmni =
mni+1 −mni
4x , Dtm
n
i =
mn+1i −mni
4t
Mxmni =
mni+1 +mni
2 , Mtm
n
i =
mn+1i +mni
2 ,
then the box scheme for (4.34) reads
DtMxm+MtMxu ·DxMtm + 2DxMtu ·MtMxm = 0
1
24t

 1 1
−1 −1

m = −14

 1 1
1 1

u · 124x

 −1 1
−1 1

m
− 12

 1 1
1 1

m · 124x

 −1 1
−1 1

u.
(4.35)
x
t
mni
mn+1i m
n+1
i+1
mni+1
Figure 4.2.1: The box scheme for (4.34)
This is an implicit scheme and we can solve the obtained algebraic equa-
tions by Newton iteration method. The result for the Gaussian initial value
is shown in the Figure 4.2.2. Here the equation (4.1) is solved by the box
66
CHAPTER 4. NUMERICS 4.3. CONCLUSIONS
scheme with a moving frame, ie, we use u−maxx(u) as our velocity function
u in the simulation and concentrate on region where the blob is mainly sup-
ported. The region is [0, L] = [0, 16], with n = 400 grid points, dt = 0.01,
and the initial value is m0(x) = e−|x−8|
2.
The graph indicates that the Gaussian initial value does evolves to a
Dirac δ function as t→ +∞ although we know that the mCH equation has
no finite time blowup solution. This directly leads to the study we do in
Chapter 5.
6 8 10 12 14 16 18
0
1
2
3
4
x
m
Blow up in the Euler equations for (Diff(R),H2)
Figure 4.2.2: The evolution of Gaussian initial value m0(x) = e−|x−8|
2.
Remark Zhao and Qin [110] proved that the box scheme is symplectic
and multi-symplectic for KdV equation, this is because KdV has another
Hamiltonian (Poisson) structure ∂x with constant coefficient. D. Cohen et
al. [24] proposed a box scheme for the Camassa-Holm equation which they
proved is multi-symplectic, this is also because the CH equation has a con-
stant Hamiltonian (Poisson) structure ∂x(1 − ∂2x) (see the last part of the
section 3.1.2). But for the general mCH, we have only one Hamiltonian
structure m∂x + ∂xm, which leads to no easy way to get a Hamiltonian
discretization in space. Actually, we can propose a multi-symplectic formal-
ization for the mCH equation in the Appendix B via the multi-symplectic
geometry approach, but it is in the Lagrangian coordinates and is not of
much practical use.
4.3 Conclusions
We have studied the numerical aspect of the limiting mCH, ie, a = 0, on
the circle S and on the whole real line R1 respectively in this chapter. We
have proved the convergence of the so-called particle methods in Section
4.1, which is quite similar to that in [17]. But here in our case, the global
67
CHAPTER 4. NUMERICS 4.3. CONCLUSIONS
existence of the corresponding ODEs is very easy due to the nice property
of the mCH with k ≥ 2, and we do not need to use the integrability which
the mCH probably does not have. Then we have proposed a box-scheme for
(4.1). This scheme is locally a second order method because it is a symmetric
method, and it is not too expensive to implement (Newton’s iteration will
solve the resulting algebraic equations). The simulation we have done gives a
reliable result in a not-very-long time. However, the nonlinearity, especially
the nonlocal dependence of u on m, make it very hard to rigorously prove
the convergence of the box-scheme.
68
Chapter 5
Asymptotics
[Paradoxes of the infinite arise] only when we attempt, with our
finite minds, to discuss the infinite, assigning to it those properties
which we give to the finite and limited.
— Galileo Galilei
5.1 Introduction
In this chapter, we are going to study the asymptotics for the limiting mCH
equation on R1, that is,



mt = −umx − 2mux on R1,
m(x, 0) = m0(x).
(5.1)
where (1− ∂2x)ku = m, k ≥ 2. As we mentioned before, this is a generalised
Euler equation on Diff(R) with respect to Hk metric. We have proved in
Chapter 3 that the solution of this equation does not blowup in any finite
time, unlike the Camassa-Holm equation. However, our numerical simulation
shows that the solution m does tend to some weak soliton-like solution as the
time t→ +∞ for some initial values (see Figure 4.2.2 in Chapter 4 showing
that the initial Gaussian value evolves with linear growth in height towards
a δ like function).
This is sort of similar to the phenomenon in Camassa-Holm equation (see
[51], where they depicted the solutions of the CH equation on the circle with
some Gaussian initial value that evolves into an ordered soliton-train as the
time t increases). We have not yet found any study on this phenomenon in
the existing literature.
69
CHAPTER 5. ASYMPTOTICS 5.2. ASYMPTOTIC PDE
First, we are going to use the asymptotic expansion to get an ODE which
the asymptotic profile satisfies, then use the so-called matched asymptotic
expansion method [105] to analyse the asymptotic behaviour for the gener-
alised Euler equation (5.1) as t goes to infinity.
The method of matched asymptotic expansions is quite powerful dealing
with two scale problems. Roughly speaking, it approximates the problem
in two separate coordinates, one in the fine coordinate and the other in the
coarse coordinate, and matches the boundary conditions using the matching
principle.
In general, if we know that an evolutionary PDE has a solution v(x, t)
which blows up at time T and we are to study the (asymptotic) self-similar
blowup profile f for the solution v(x, t) of the PDE, we need to choose an
appropriate “similarity variable ξ”, which may depends on x and t, and a
scaled factor φ(t) (may depending on T ) according to the nature of PDE
such that when plugging the solution v(x, t) = φ(t)f(ξ) into the PDE, we
can obtained a differential equation of f as t goes to the blowup time T.
The travelling wave solutions of PDEs are closely related to the self-similar
solutions which Barenblatt [7] discussed intensively for partial differential
equations. Here by the term “travelling wave” we mean the solutions of
the form f(x − ct). If one is looking for travelling wave solutions u(t, x) =
f(x − ct), then he can regard ξ = x − ct as a new variable and plug this
ansatz u(t, x) = f(x− ct) back into the PDE to get an ODE in ξ.
For some PDEs, the resulted differential equation on f has a unique
(stable) solution, e.g. [80]-[83] for the generalised KdV equations and [106] for
travelling wave solutions of parabolic systems, and we can use various tools
to prove that the solution of the PDE tends to the unique steady solution in
some sense.
But for the mCH equation, we will find in this chapter that the situation
is quite different.
5.2 Asymptotic PDE
We can observe from Figure 4.2.2 that the solution is getting taller and
thinner as the time t increases and the bump moves to the right at an almost
constant speed. This observation is helpful although it is not necessary to
our analysis. In order to study how the solutions are approaching the blowup
profile, we consider the travelling wave solutions of the form
m(t, x) = φ(t)f(φ(t) · (x− ct))
70
CHAPTER 5. ASYMPTOTICS 5.2. ASYMPTOTIC PDE
with a scaling factor φ(t) (we take this form to guarantee
∫
m =
∫
f). We
will first plug this ansatz into the equation (5.1) to find the right choice of
φ(t), and then get the differential equation that f satisfies. If G denotes the
Green’s function for the operator (1− ∂2x)k, then for very large φ = φ(t), we
can approximate u by
u(y) =
∫
G(y − x)m(x)dx
=
∫
G(y − x)φf(φ(x− ct))dx
=
∫
G(y − ξ/φ− ct)f(ξ)dξ ξ = φ(t) · (x− ct)
=
∫
G(η − ξφ )f(ξ)dξ η = φ(t) · (y − ct)
=
∫ (
G0 +
(η − ξ)2
2φ2 G
′′(0)
)
f(ξ)dξ +O( 1φ3 ) with G0 = G(0)
= G0f0 +
G′′(0)
2φ2 f2 +
G′′(0)
2φ2 f0η
2 − ηφ2G
′′(0)f1 +O(
1
φ3 ),
(5.2)
where fi =
∫
ξif(ξ)dξ. Here we have used the fact G′(0) = 0. From (5.2),
we have for very large φ that
uy =
∫
G′y(y − x)m(x)dx
=
∫ G′′(0)(η − ξ)
φ f(ξ)dξ +O(
1
φ2 )
= G′′(0)(ηf0 − f1)/φ+O(
1
φ2 )
(5.3)
Substituting (5.2)(5.3) into (5.1), we get a differential equation:
φf ′ · (φ′η/φ− cφ) + φ′f + 2G′′(0)(ηf0 − f1)f +
+
(
G0f0 +
G′′(0)
2φ2 f2 +
G′′(0)
2φ2 η
2f0 −
G′′(0)
φ2 ηf1
)
φ2f ′ = 0.
(5.4)
ie
ηf ′φ′−cf ′φ2+fφ′+G0f0f ′φ2+2G′′(0)(ηf0−f1)f+
G′′(0)
2
(
f2 + η2f0 − 2ηf1
)
f ′ = 0.
(5.5)
71
CHAPTER 5. ASYMPTOTICS 5.2. ASYMPTOTIC PDE
In order to make this equation balance in φ, we have to assume φ′ ∼ 1 or
φ′ ∼ φ2. We will choose which is suitable to our goal.
 If φ′ ∼ φ2, take φ′ = φ2 as an example, then φ(t) will become infinity
at some finite time, which contradicts with our result in Chapter 3.
Moreover, if φ′ = φ2, then the leading term in (5.5) will lead to
ηf ′ − cf ′ + f +G0f0f ′ = 0
which has only unbounded solutions. This is not of interest to us
because we are looking for some smooth profile. This rules out the
choice φ′ ∼ φ2.
 If φ′ ∼ 1, take φ′ = 1 for example, we have φ(t) = t and we can get the
case in which we are interested.
Now if we take φ(t) = t. Then we match the coefficients of ti
t0 : [2G′′(0)(ηf0 − f1) + 1] f
+
(G′′(0)
2 f2 +
G′′(0)
2 η
2f0 −G′′(0)ηf1 + η
)
f ′ = 0,
t2 = φ2 : −cf ′ +G0f0f ′ = 0.
The leading order term t2(−cf ′ +G0f0f ′) is the limiting (δ function) part of
the motion. If we divide the equation (5.4) by t2 and let t→∞, then we get
−cf ′+G0f0f ′ = 0, which means the leading order term determines the speed
c of the soliton: c = G0f0. If we denote U(η) = 12G′′(0)f0η2 − G′′(0)f1η +1
2G′′(0)f2, then the ODE for the coefficients in t0 takes the form
(ηf)η + fηU + 2Uηf = 0. (5.6)
This equation is nonlinear and nonlocal (U depends on the integrals of
f), but fortunately it can be explicitly solved as follows:
Multiplying the LHS of the ODE in t0 by η and then integrating it, we
find that f1 = 0. Because G′′(0) < 0, we assume f0G′′(0) = −1, f2G′′(0) =
−1− a−2, then we have
(−2η + 1)f + 12(2η − η
2 − 1− a−2) · f ′ = 0, (5.7)
so
df
f =
−2(2η − 1)dη
(η − 1)2 + a−2
72
CHAPTER 5. ASYMPTOTICS 5.2. ASYMPTOTIC PDE
then
−dff =
2 + 4(η − 1)
(η − 1)2 + a−2 dη =
2
(η − 1)2 + a−2 dη +
4(η − 1)
(η − 1)2 + a−2 dη.
From which we have
f(a, η) = C(a) · e
−2a[arctan(a(η−1))+pi2 ]
((η − 1)2 + a−2)2 . (5.8)
We can find, with the help of the software Mathematica, that
∫ +∞
−∞
fdη = C(a)a
2
4
1− e−2api
a2 + 1 ,
∫ +∞
−∞
η2fdη = C(a)4 (1− e
−2api).
If we take
C(a) = − 1G′′(0)
a2 + 1
a2 ·
4
1− e−2api
then f satisfies the conditions f0G′′(0) = −1, f2G′′(0) = −1−a−2. The limit
of (5.8) is
lim
a→+∞
f(a, η) = f˜ =



4(η − 1)−4e 2η−1 if η − 1 < 0,
0, if η − 1 > 0.
(5.9)
Here in (5.8), we have taken an integral constant C(a)e−api in order to guar-
antee the limit function f˜ ∈ L1(R). It’s easy to verify that f˜ is a solution
to (5.6) too, and we call this solution the limit steady solution. The steady
solutions are depicted in Figure 5.2.1
Remark There are two magic features of the equation (5.6).
 The equation (5.6) is a nonlinear differential equation, so in general we
can not multiply or divide the solution by a constant to get a solution,
which means the assumption f0G′′(0) = −1 is not very plausible, be-
cause when we assume f0G′′(0) = −1 and f2G′′(0) = −1 − a−2, then
the equation becomes a linear equation and we can solve it without any
difficulty. But the check with Mathematica shows that the two a-prior
assumptions on f0 and f2 does not lead to contradiction. Perhaps, we
can think of it this way: because f0 could be any number, so we take
f0G′′(0) = −1 and think f2 as a parameter (where the parameter a
comes from).
73
CHAPTER 5. ASYMPTOTICS 5.2. ASYMPTOTIC PDE
−4 −3 −2 −1 0 1 2 3 4
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
a=2
a=3
limit case (a=∞)
ξ
f
the steady solutions  to the asymptotic equations
Figure 5.2.1: The steady solutions to the asymptotic equations
 The most interesting point here is that there is a family of self-similar
blowup profiles, unlike most familiar cases where just a unique pro-
file exists [80]-[83], [106]. This is the first equation with this kind of
property we have ever seen.
If we take m(x, t) = tf(t(x − ct), τ), with a time scale factor τ = g(t),
then we can do the same things as before
tfξ · (η/t− ct) + f + tfτg′(t) + 2G′′(0)(ηf0 − f1)f
+
(
G0f0 +
G′′(0)
2t2 f2 +
G′′(0)
2t2 η
2f0 −
G′′(0)
t2 ηf1
)
t2fξ = 0.
(5.10)
This means that if we want to take fτ into the equation, we have to take
g′(t) = 1t , that is g(t) = ln t. Then we have got the following asymptotic slow
time PDE (the self-similar case) for which the equation (5.7) is the stationary
equation:
fτ + (ξf)ξ + Ufξ + 2Uξf = 0, (5.11)
here U = G′′(0)2 (ξ2f0 − 2ξf1 + f2). It is easy to verify that f0 is independent
of t, which we can assume satisfies f0G′′(0) = −1 in order to simplify the
calculation.
The numerical simulation strongly suggests that the steady solutions (5.8)
and (5.9) are stable for the asymptotic equations (5.11) and are stable asymp-
totical solutions to the equation (5.1)(see Figures 5.4.1— 5.4.3) (here we use
the term “stable asymptotical solutions” to mean the solutions of (5.1) tends
74
CHAPTER 5. ASYMPTOTICS 5.3. ARE STEADY SOLUTIONS STABLE?
to the family of “steady solutions” (5.8) and (5.9), but we have to point out
that (5.8) and (5.9) are not the steady solutions of (5.1), although they are
the steady solutions of (5.11) ). We will give some trials to studying this
stability in the next section.
Remark Our analysis applies to all Hk metric with k ≥ 2, but not
directly to H1 metric which corresponds to the Camassa-Holm equation be-
cause in this case G′(0), G′′(0) do not exist at all. We will have a look at
the Camassa-Holm equation in the last section of this chapter.
5.3 Are Steady Solutions Stable?
In order to study the stability of the steady solution f, f as in (5.8)(5.9), we
linearise the equation (5.11) around this steady f by a small perturbation
f + εh(τ, ξ) and get the linearised equation on h
hτ + [(U + ξ)h]ξ + Uξh+ (Wf)ξ +Wξf = 0, (5.12)
where U = G′′(0)2 (f0ξ2+f2), W =
G′′(0)
2 (h0ξ2−2h1ξ+h2) and hi =
∫
ξih(ξ)dξ.
One can easily find that
dh0
dτ = 0,
dh1
dτ = h1,
dh2
dτ +
G′′(0)
2 (h0f + f0h)ξ
4 ∣∣+∞
−∞ = 2h2. (5.13)
In order to study the stability of f , the natural idea is to prove that h(τ, ξ)
stays small in some sense as τ → +∞ for small initial value h(0, ξ).
We are trying to study the stability of the steady solutions by charac-
teristic methods. One of the difficulties is that one can easily see that some
characteristics of (5.12) can not reach the ξ axis(ie, τ = 0) (see Figure 5.3.3
and Figure 5.3.5 below) and we need some upstream boundary conditions
h(τ, ξ0) for some large ξ0.
5.3.1 Upstream Boundary Conditions for (5.12)
In the previous section, we let ξ = t(x − ct), with c = G(0)f0, as a new
variable, which means we concentrate on the region where the bump is sup-
ported and which is getting smaller and smaller as t increases. Now we are
going to obtain some upstream boundary conditions for the equations (5.11)
and (5.12), that is the boundary condition for ξ very large, that is where the
so-called matched asymptotic expansion method comes in.
The technique of matched asymptotic expansion concerns different dif-
ferential equations in two regions known as the “fine scale” (or inner) region
75
CHAPTER 5. ASYMPTOTICS 5.3. ARE STEADY SOLUTIONS STABLE?
and the “coarse scale” (or outer) region, matching the boundary condition
on the “common” boundary (see [7] and [105] for example). The multi-scale
asymptotic expansion is very subtle. It appears like “what you get depends
on what you want”. The key point here is the scale under which the prob-
lem is considered, or in other words, what limit process is of interest to us.
Specific to our problem here, we can think the region A with the variable
ξ = t(x − ct) as the inner region (with ξ = O(1)) and think the region
out of A as an outer region with another variable much coarse, for example,
z = x − ct = O(1). Now we need the upper stream boundary condition for
f(ξ, τ) or h(ξ, τ), we can think this as the boundary condition at z → 0+ of
the outer region equation as t→ +∞ (see Figure 5.3.1).
the inner region : ξ=t(x−ct) coordinate
the outer region: z=x−ct coordinate
The inner region and the outer region
Figure 5.3.1: The illustration of the inner and outer regions. Take t = 5 for
example, and there are two points P1, P2, if the distance of P1, P2 in the ξ
coordinate is dξ(P1, P2) = 10, then the distance of these two points in the z
coordinate is dz(P1, P2) = 10/5 = 2. Here we plot a function f(ξ) on the top
and the function m(t, z) = tf(ξ) on the bottom
Now we let z = x−ct as a new variable, and have another approximation
for u(x) in (5.1) (when t is very large, while we think z = x − ct = O(1),
76
CHAPTER 5. ASYMPTOTICS 5.3. ARE STEADY SOLUTIONS STABLE?
because now we are in the z coordinate):
u(y) =
∫
G(y − x)m(x)dx
=
∫
G(y − ct− ξt )f(ξ)dξ,
if m(x) = tf(t(x− ct)), ξ = t(x− ct), c = G0f0,
≈ G(y − ct)f0 −
1
t · · ·
(5.14)
so in a moving frame (which means we can let z = x− ct denote the relative
position of the point x with respect to the point where the Dirac δ function
is supported, and consider m as a function of z : m(t, x) = n(t, z) = n(t, x−
ct)), then m(0, x) = n(0, z) = n(0, x) and mt = nt−cnz. If we denote v(z) =
(G(z) − G0)f0, then v(z) = u(x) − c, vz = ux and we can approximate the
equation (5.1) (in the region where t is large enough and z = x− ct = O(1))
by
nt + vnz + 2vzn = 0. (5.15)
If we take f0 = 1 just for simplifying the notations, then v(z) = (G(z) −
G0)f0 = G(z)− G0, and this linear equation can be explicitly solved by the
characteristic method:
n(t, z) = n(0, F (−t, z))
(G0 −G(F (−t, z))
G0 −G(z)
)2
= m(0, F (−t, z))
(G0 −G(F (−t, z))
G0 −G(z)
)2
= m0(F (−t, z))
(G0 −G(F (−t, z))
G0 −G(z)
)2
,
(5.16)
where z = F (t, z0) is defined by the characteristics curve
∫ z
a
1
G(x′)−G0
dx′ = t+
∫ z0
a
1
G(x′)−G0
dx′ (5.17)
where
a =



1 if z0 > 0,
−1 if z0 < 0.
77
CHAPTER 5. ASYMPTOTICS 5.3. ARE STEADY SOLUTIONS STABLE?
0 0.5 1 1.5 2
−60
−40
−20
0
20
40
60
The graph of the function f(x) = −
∫ x
1
1
G(x)−G0
dx
↑ α+ xG0
← 2G”(0)x
x
Figure 5.3.2: Asymptotic for the function f(z) ≡ −
∫ z
1
1
G(x′)−G0
dx′
z = F (t, z0) means the curve starting from z0 at t = 0 arrives at z at the
time t, so it is clear that z0 = F (−t, z).
Now we are going to use the limit of m(t, z) as t → +∞ and z → 0+ as
the boundary condition of the outer region with (z, t) coordinates and then
transfer this limit to the upstream boundary condition of the inner region
with (ξ, τ) coordinates.
In order to do that, we need to find the asymptotic expression of m(t, z)
as z > 0 very small and t large enough, ie, we are interested in the case
z → 0+, and t → ∞, such that ξ = zt = t(x − ct) > 0 very large. We
are concerned with how fast the initial value m0(z0) decreases as z0 → +∞
can guarantee the upstream boundary conditions of f(ξ, τ) tends to 0 as
τ → +∞, so we consider the situation where z0 → +∞, z → 0+, t→ +∞.
In this case, taking into account that G(z) = G0 + 12G′′(0)z2 +o(z3) for z > 0
small enough and G(z)→ 0 as z → +∞, we have
−
∫ z
1
1
G(x′)−G0
dx′ ≈ 2G′′(0)z is negatively large as z > 0 small enough
−
∫ z0
1
1
G(x′)−G0
dx′ ≈
(
α+ z0G0
)
is positively large as z0 large enough,
78
CHAPTER 5. ASYMPTOTICS 5.3. ARE STEADY SOLUTIONS STABLE?
(see Figure 5.3.2). On the other hand, from (5.17), we have
−
∫ z0
1
1
G(x′)−G0
dx′ ≈ 1z
(
tz + 2G′′(0)
)
,
and so z0 ≈ G0z (tz + 2G′′(0)), and
m(t, x) = n(t, z) = m0(z0)
(G0 −G(F (−t, z))
G0 −G(z)
)2
≈ G
2
0
(G0 −G(z))2
m0
(G0
z (tz +
2
G′′(0))
)
.
(5.18)
This approximation holds true for t large enough and z > 0 small enough.
Now we will match thism(t, z) to the upstream boundary condition of f(ξ, τ).
Notice G(z) − G(0) = G′′(0)2 z2 + o(z3) as z → 0+, so the estimate (5.18),
together with m = tf(ξ, τ), τ = ln t, ξ = zt, yields that for large ξ and τ ,
f(ξ, τ) = e3τξ−4( 2G0G′′(0))
2m0
(
G0eτ + 2
G0eτ
G′′(0)ξ
)
. (5.19)
This can give a f(ξ, τ) such that lim
τ→∞
f(ξ, τ) = 0 as long as the initialm0(x) =
o(x−3) as x → +∞. We call this f(ξ, τ) for large ξ = ξ0 the upstream
boundary condition for the inner problem (5.11). So the perturbation h(τ, ξ0)
can be assumed to be h(τ, ξ) = O(1) · f(ξ0, τ) which we call the upstream
boundary condition for the inner problem (5.12). The above process is the
so-called the matched asymptotic expansion method.
5.3.2 Around the General Steady Solution (5.8)
We first consider the general steady solution (5.8) case. In this case, the
problem is that the characteristics of (5.12) are like arctan(ξ) (see Figure
5.3.3 ) and the initial value of h(τ, ξ) is not the initial h(0, ξ), and instead,
is determined by some boundary conditions h(τ0, ξ0) for some large ξ0 and
τ0 > 0.
In fact, the characteristic is determined by
dξ
dτ = U + ξ, (5.20)
and if we put G′′(0)f0 = −1, G′′(0)f2 = −1− a−2, then this equation reads
dξ
dτ = −
1
2
[
(ξ − 1)2 + a−2
]
, (5.21)
79
CHAPTER 5. ASYMPTOTICS 5.3. ARE STEADY SOLUTIONS STABLE?
−10 0 10 20 30 40
−6
−4
−2
0
2
4
6
ξ
t
Characteristics of the linearised equation around the general steady solutions
h(τ,ξ 0) given
Figure 5.3.3: Characteristics of the linearised equations around the general
steady solutions
and the characteristic curves are
arctan a(ξ − 1) = − 12aτ +K, (5.22)
where K is determined by the upstream boundary, ie,
K = 12aτ0 + arctan(a(ξ0 − 1)). (5.23)
Along the characteristics, the equation (5.12) reads
dh
dτ + [1 + 2Uξ]h+ (Wf)ξ +Wξf = 0. (5.24)
For any ξ1 > 0 fixed and any τ1 large, there is a unique characteristic connect-
ing (ξ1, τ1) and (ξ0, τ0) for some τ0, and |τ1−τ0| < api, where a is the parameter
from G′′(0)f2 = −1 − a−2. Let A = max{|1− 2ξ| : ξ between ξ1 and ξ0}, A
is large but fixed once ξ0, ξ1 fixed, then from
h(τ1, ξ1) = e−
∫ τ1
τ0
(1−2ξ)
[
h(τ0, ξ0) +
∫ τ1
τ0
(−2Wξf −Wfξ)e
∫
(1−2ξ)
]
, (5.25)
if h(τ0, ξ0)→ 0 as τ0 → +∞ and W, Wξ small(which means from (5.13) that
h0, h2 small and h1 = 0), we have
lim sup
τ1→∞
h(τ1, ξ1) is small. (5.26)
80
CHAPTER 5. ASYMPTOTICS 5.3. ARE STEADY SOLUTIONS STABLE?
If we take h0 = h1 = 0 and G′′(0)f0 = −1, then (5.13) reads
dh2
dτ −
1
2c(τ) +
1
2hξ
4|−∞ = 2h2, (5.27)
where c(τ) = lim
ξ→+∞
hξ4 and lim
τ→∞
c(τ) = 0. From (5.25) we have
h(τ1, ξ1) = e−
∫ τ1
τ0
(1−2ξ)
[
h(τ0, ξ0)−
G′′(0)
2
∫ τ1
τ0
fξh2e
∫ τ1
τ0
(1−2ξ)
]
≡ I + II.
(5.28)
But along the characteristics, we have
e−
∫ τ1
τ0
(1−2ξ) = eτ1−τ0
[1 + a2(ξ0 − 1)2
1 + a2(ξ1 − 1)2
]2
, (5.29)
and so
I = e−
∫ τ1
τ0
(1−2ξ)h(τ0, ξ0)
= eτ1−τ0
[1 + a2(ξ0 − 1)2
1 + a2(ξ1 − 1)2
]2
h(τ0, ξ0)
= eτ1−τ0
[ 1
1 + a2(ξ1 − 1)2
]2
c(τ0)a4
(5.30)
because h(τ0, ξ0) ≈ c(τ0)ξ40 . If we let τ1 fixed, then τ0 = τ0(ξ1) depends on ξ1.
lim
ξ1→−∞
I · ξ41 = limξ1→−∞ e
τ1−τ0c(τ0) = e2apic(τ1 − 2api) (5.31)
for τ0 = τ1 − 2a arctan(a(ξ0 − 1)) + 2a arctan(a(ξ1 − 1)). On the other hand,
fξ =
−4ξ
a−2 + (ξ − 1)2f, (5.32)
so
II = 2G′′(0)eτ1−τ0 1[a−2 + (ξ1 − 1)2]2
∫ τ1
τ0
h2fξeτ0−τ (a−2 +(ξ−1)2)dτ. (5.33)
Plugging the formula of f and the characteristic equation (5.22), then a direct
81
CHAPTER 5. ASYMPTOTICS 5.3. ARE STEADY SOLUTIONS STABLE?
calculation yields
lim
ξ1→−∞
II · ξ41
= 2G′′(0)eτ1−api lim
ξ1→−∞
∫ τ1
τ0
h2e−2aK
1
a tan(K − 12aτ) + 1
a−2 tan2(K − 12aτ) + a−2
dτ
= 2aG′′(0)eτ1−api lim
ξ1→−∞
∫ τ1
τ0
h2e−2aK
[
a cos2(K − 12aτ)
+ cos(K − 12aτ) sin(K −
1
2aτ)
]
dτ
= 2aG′′(0)
∫ τ1
τ1−2api
[ah2
2
{
1 + cos
(τ1 − τ
a − pi
)]
+12h2 sin
(τ1 − τ
a − pi
)}
dτ,
(5.34)
because
lim
ξ1→−∞
K = lim
ξ1→−∞
1
2aτ0 +arctan(aξ0−a) =
1
2aτ1−pi+
pi
2 =
1
2aτ1−
pi
2 . (5.35)
So
lim
ξ1→−∞
II · ξ41
= 2aG′′(0)
∫ τ1
τ1−2api
[ah2(τ)
2
(
1− cos
(τ1 − τ
a
))
−12h2(τ) sin
(τ1 − τ
a
)]
dτ,
(5.36)
so h2 satisfies at τ = τ1
dh2
dτ −
1
2c(τ1) +
1
2c(τ1 − 2api)e
2api − 2h2
= −12aG
′′(0)
∫ τ1
τ1−2api
[
a
(
1− cos
(
τ1 − τ
a
))
− sin
(
τ1 − τ
a
)]
h2dτ.
(5.37)
If we can obtain from this differential integral equation that h2(τ) stays small
for all large time τ , then it is easy to see from (5.25)
lim
τ→+∞
h(τ, ξ) = 0. (5.38)
But the problem here is that we can not have the smallness of h2 from (5.37)!
In fact, we can numerically solve the equation (5.37) and the result is shown in
82
CHAPTER 5. ASYMPTOTICS 5.3. ARE STEADY SOLUTIONS STABLE?
the following Figure 5.3.4, which indicates that h2(τ) increases exponentially
as a function of τ . (In this figure, we take c(τ) ≡ 0, a = 1, G′′(0) = −1 and
h2(τ) = −0.1 for τ ≤ 0.) However, this example does not contradict with
(5.38) although we can not derive any definite result from it.
0 2pi 4pi 6pi 8pi 10pi
10−5
100
105
1010
1015
1020
1025
1030
semilogy plot of  h2 
τ
log
(h 2)
Figure 5.3.4: Plot of h2(τ)
5.3.3 Around the Limit Steady Solution (5.9)
At this stage, the characteristic equation of (5.12) is
dξ
dτ = −
1
2(ξ − 1)
2 (5.39)
and the characteristics from ξ|τ=τ0 = ξ0 are
(ξ − 1)−1 = 12(τ − τ0) + (ξ0 − 1)
−1
whose diagrams are shown in Figure 5.3.5. Clearly, τ − τ0 < (ξ − 1)−1 for
ξ > 1.
Along each characteristic curve, the linearised equation (5.12) becomes
dh
dτ + (1 + 2Uξ)h+ (Wf)ξ +Wξf = 0, (5.40)
83
CHAPTER 5. ASYMPTOTICS 5.3. ARE STEADY SOLUTIONS STABLE?
−0.5 0 0.5 1 1.5 2 2.5 3
−250
−200
−150
−100
−50
0
50
ξ
t
ξ =1
Characteristics of the linearised equation around the limit steady state
Figure 5.3.5: Characteristics of the linearised equations around the limit
steady solution
ie
dh
dτ + (1− 2ξ)h+ (Wf)ξ +Wξf = 0. (5.41)
Case ξ > 1.
For any ξ > 1, we have f ≡ 0, so this equation reads
dh
dτ = (2ξ − 1)h (5.42)
whose solution can be found explicitly
h(τ, ξ) = eτ−τ0
(ξ0 − 1
ξ − 1
)4
h(τ0, ξ0). (5.43)
If ξ > 1 is fixed, then τ0 → +∞ as τ → +∞ because τ − τ0 < (ξ − 1)−1, so
eτ−τ0 is bounded, from which we have for fast decaying h(τ, ξ0)
lim
τ→∞
h(τ, ξ) = 0 for any ξ > 1.
Case ξ < 1.
If ξ < 1, then for any τ > 0 there is a characteristic curve connecting (ξ, τ)
and (ξ0, 0) for some unique ξ0 < 1. Along each such curve, the equation (5.12)
reads
dh
dτ + (1− 2ξ)h = −(Wf)ξ −Wξf. (5.44)
84
CHAPTER 5. ASYMPTOTICS 5.4. THE FAMILY OF STEADY SOLUTIONS
Denote l(ξ) = −(Wf)ξ −Wξf, then l(ξ) = o(1) · (ξ − 1)−4e
2
ξ−1 (ξ − 1)−2 if
W, Wξ = o(1). Along the characteristic curve, we have
e−
∫ τ
0 (1−2ξ) = e
2
ξ−1−
2
ξ0−1
(ξ0 − 1
ξ − 1
)4
(5.45)
and the solution is
h(τ, ξ) = e−
∫ τ
0 (1−2ξ)
[
h(0, ξ0) +
∫ τ
0
l(ξ(s))e
∫ s
0 (1−2ξ)ds
]
. (5.46)
the first part of the solution
e
2
ξ−1−
2
ξ0−1
(ξ0 − 1
ξ − 1
)4
h(0, ξ0) (5.47)
can be small if h(0, ξ0) = o(1) · e
2
ξ0−1 (ξ0 − 1)−4 = o(1) · f˜(ξ0).
But the other part
e−
∫ τ
0 (1−2ξ)
∫ τ
0
l(ξ(s))e
∫ s
0 (1−2ξ)ds = e 2ξ−1
( 1
ξ − 1
)4
· o(1) ·
∫ τ
0
(ξ(s)− 1)−2ds
(5.48)
is not small! This means that we can not use the standard approach to
establish the linear stability of the limit steady solution.
5.4 The Family of Steady Solutions
Then we go further with the numerical simulation on the stability of steady
solutions. We found that the real profile seems to wander within the family
of steady solutions, as shown in the Figures 5.4.1– 5.4.3. In these figures,
the differential equation (5.1) is solved by the box scheme with a moving
frame, ie, we use u − maxx(u) as our velocity function u in the simulation
and concentrate on region where the blob is mainly supported. The region
is [0, L] = [0, 12], with n = 400 grid points, dt = 0.01, and the initial value is
m0(x) = e−|x|2. The solid line of the top plot of each figure stands for the real
solution m(x, t), and then use the nonlinear least square method to solve a
parameter optimization problem to find the closest profile (hence the shape
parameter a) from the family
a3ea4
exp(−2a2 arctan(a2a3(x− a1)))
((a3(x− a1))2 + a−22 )2
, (5.49)
85
CHAPTER 5. ASYMPTOTICS 5.4. THE FAMILY OF STEADY SOLUTIONS
with four parameters ai, i = 1, · · · , 4: a1 is the maximum point, a2 is the
shape parameter a, a3 is the spatial scaling factor and a4 the height scaling
factor. The solid line in the figures is the numerical solution of the equation
(5.1) and the dotted line correspond to the asymptotic profile selected from
(eq:ft) that best fits the solution. We can see from the Figure 4.2.2 and the
Figures 5.4.1– 5.4.3 that the Gaussian initial value evolves very quickly to
the family of steady solution profiles, then it does not stay at any steady
solution, instead, it wanders in this family of steady solutions (see the first
plot of the evolution of a in Figure 5.4.4 ). It is so strange that we have not
ever seen the similar phenomenon before.
Here are some excuses why we failed to rigorously solve this problem:
 Analytically, both the original PDE (5.1) and its slow time asymptotic
PDE (5.11) are nonlinear and nonlocal, and are not real hyperbolic
equations. Normally, when talking about the stability of a steady so-
lution, we need some nice properties such as the dissipation or some
enough number of conserved quantities to guarantee the solutions of the
PDE tends to the steady solution in some sense, of which the PDEs
(5.1)(5.11) lack.
 Numerically, we can not reliably solve the asymptotic PDE (5.11) be-
cause of the coefficients in (5.11) depends on fi, i = 0, 1, 2. When we
solve numerically (5.11), after some time, the solution f(η, τ) behaves
like (5.8) or (5.9), which means
f ∼ η−4 for |η| large
which incurs an O( 1L) error when we replace f2 =
∫∞
−∞ ξ2f(ξ)dξ with
the evaluation of f2 =
∫ L
−L ξ2f(ξ)dξ in the simulation. This error will
take over the true solution, so we are not able to check the stability
numerically.
 We can not solve the mCH reliably for long times because of the weak
blowup. If we solve directly the original equation (5.1), then the solu-
tion will be getting larger and larger as the time increases due to the
weak blowup.
 If we use a moving frame in the simulation, then we concentrate on the
very narrow region where the bump is supported, which is good. But
a moving frame means that we are not capturing the correct upstream
boundary conditions. It seems that it would need the adaptive grid
method used by Budd [16] which we could not get working.
86
CHAPTER 5. ASYMPTOTICS 5.4. THE FAMILY OF STEADY SOLUTIONS
 The reason that the standard characteristic method does not work is
probably that the characteristic curves depend on the solutions, which
means that, as the solutions of (5.1) tend to (5.8)(5.9), the character-
istic curves themselves change!
From the numerical simulations, we make the following conjectures:
 The one parameter family of the steady solutions is exponentially sta-
ble, every initial value not in but close to this set will tend to it very
quickly and then wanders along this set.
 If the initial value m(x, 0) is zero for all large x > 0, then the solution
will tend to the limit steady solution (5.9).
 If m(x, 0) is zero for all large x > 0 and there is a small perturbation
h(x0, t) at some large x0 with h(x0, t)→ 0 as t→∞, then the solution
will track this perturbation and tends to the limit steady solution (5.9)
as t→∞.
87
CHAPTER 5. ASYMPTOTICS 5.4. THE FAMILY OF STEADY SOLUTIONS
0 2 4 6 8 10 12
0
0.5
1
1.5
shape parameter a = 0.12
time t = 0.00
x
m
−4 −3 −2 −1 0 1 2
0
0.5
1
1.5
η
sc
ale
d m
0 2 4 6 8 10 12
−0.5
0
0.5
1
1.5
shape parameter a = 1.37
time t = 4.00
x
m
−4 −3 −2 −1 0 1 2
0
0.5
1
1.5
η
sc
ale
d m
Figure 5.4.1: Fit of the solution to the true profiles: the equation is solved by
the box scheme with a moving frame. Starting off from the Gaussian initial
value, the solution is soon almost indistinguishable from the true profile.
88
CHAPTER 5. ASYMPTOTICS 5.4. THE FAMILY OF STEADY SOLUTIONS
0 2 4 6 8 10 12
−0.5
0
0.5
1
1.5
2
shape parameter a = 2.13
time t = 8.00
x
m
−4 −3 −2 −1 0 1 2
0
0.5
1
1.5
η
sc
ale
d m
0 2 4 6 8 10 12
0
1
2
3
shape parameter a = 2.17
time t = 12.00
x
m
−4 −3 −2 −1 0 1 2
0
0.5
1
1.5
η
sc
ale
d m
Figure 5.4.2: Fit of the solution to the true profiles
89
CHAPTER 5. ASYMPTOTICS 5.4. THE FAMILY OF STEADY SOLUTIONS
0 2 4 6 8 10 12
0
1
2
3
4
5
shape parameter a = 2.00
time t = 18.00
x
m
−4 −3 −2 −1 0 1 2
0
0.5
1
1.5
η
sc
ale
d m
0 2 4 6 8 10 12
0
2
4
6
8
shape parameter a = 1.90
time t = 26.00
x
m
−4 −3 −2 −1 0 1 2
0
0.5
1
1.5
η
sc
ale
d m
Figure 5.4.3: Fit of the solution to the true profiles: the solution is getting
concentrated on a narrower region as t increases, and the graph starts to
wiggle as can be seen from the plot of the scaled m.
90
CHAPTER 5. ASYMPTOTICS 5.4. THE FAMILY OF STEADY SOLUTIONS
0 10 20
0
0.5
1
1.5
2
t
a
0 10 20
0
2
4
6
8
t
sp
at
ia
l s
ca
lin
g
0 10 20
0
0.02
0.04
0.06
0.08
0.1
t
|∆m
| / 
|m
|
Figure 5.4.4: The evolution of the parameters: the first plot is on the shape
parameter a; the second is on the spatial scaling factor and the last one on
the error of the fit. The spatial scaling factor increases linearly after t ≥ 10,
as we expect, and the error decreases steadily after t ≥ 10.
91
CHAPTER 5. ASYMPTOTICS 5.5. REMARKS ON THE CAMASSA-HOLM EQUATION
5.5 Remarks on the Camassa-Holm Equation
For the H1 metric case, ie the Camassa-Holm equation, G(x) = 12e−|x|, if we
let m(t, x) = φ(t)f(φ(t)(x−ct)) and plug this ansatz into the Camassa-Holm
equation, then similarly as we did in section 5.2, we can obtain φ(t) = et from
the balance of φ in the resulted equation. In fact, if we suppose m(t, x) =
φ(t)f(φ(t)(x − ct)), we know that G′(0) does not exist in this case, but we
still have
u(y) =
∫
G(y − x)m(x)dx
=
(∫
ξ<η
+
∫
ξ>η
)
G(η − ξφ )f(ξ)dξ
= G0f0 +
1
φ(t)
[
G′+(ηf−0 − f−1 ) +G′−(ηf+0 − f+1 )
]
+G
′′(0)
2φ(t)2
[
η2f0 − 2ηf1 + f2
]
+O( 1φ(t)3 ),
(5.50)
where fi as before, and G′± = limξ→±0G
′(ξ) = ∓12, f
+
i =
∫
ξ>η
ξif(ξ)dξ, f−i =
∫
ξ<η
ξif(ξ)dξ and we denote G′′(0) = G′′+(0) = G′′−(0) = 12 although G′′(0)
does not exist actually. Similarly,
uy(y) = G′+f−0 +G′−f+0 +
G′′(0)
φ(t) [ηf0 − f1] +O(
1
φ(t)2 ). (5.51)
On the other hand,
mt = φ′ · (f + ηfη)− φ2 · cfη, my = φ2 · fη. (5.52)
Plugging all these terms into (5.1), and noticing that G′+ = −G′−, we have
φ′ · (ηf)η − φ2cfη + φ2fηG0f0
+φfηG′+[η(f−0 − f+0 ) + (f+1 − f−1 )] + 2φfG′+(f−0 − f+0 )
+12G
′′(0)fη[η2f0 − 2ηf1 + f2] + 2fG′′(0)(f0η − f1) = 0.
(5.53)
In order to balance in φ in this equation, we may have three choices: φ′ ∼
φ2; φ′ ∼ 1 or φ′ ∼ φ. We can discuss them separately as we did for the H2
92
CHAPTER 5. ASYMPTOTICS 5.6. CONCLUSIONS
metric case and obtain that only φ′ ∼ φ is of interest to us. So we take
φ(t) = et and suppose
m = etf(et(x− ct)), (5.54)
then, similarly to the previous derivation, we have the asymptotic steady
equation
(ηf)η +Wfη + 2Wηf = 0, (5.55)
where
W (η) =
[
G′+ · (ηf−0 − f−1 ) +G′− · (ηf+0 − f+1 )
]
= 12[η(f
+
0 − f−0 )− (f+1 − f−1 )].
(5.56)
Unlike in the H2 metric case, this equation is really nonlinear in f because
W depends on f in a sort of complicated way. We are not so lucky any more
as with the equation (5.6) and can not find the solution explicitly at the
moment. Hopefully, we will work on this equation in the future.
5.6 Conclusions
In this chapter, we have studied the asymptotic behaviour of the mCH (5.1)
on the whole real line R1 by the asymptotic expansion and the so-called
asymptotic matching method. After a short introduction, we have used the
asymptotic expansion in Section 5.2 to derive the ODE (5.6), which the
asymptotic steady solutions should satisfy, and the slow-time evolutionary
PDE (5.11). The ODE (5.6) admits a family of solutions (5.8),(5.9). When
linearising the slow-time PDE (5.11) around the steady solutions (5.8)(5.9),
we have found that the characteristics do not intersect with the x-axis, which
means we have to assign some upstream boundary conditions for the lin-
earised equations. So we have approximated the equation (5.1) in another
(coarser) scale to get (5.15). Matching the solution of (5.15) to the upstream
boundary condition of (5.15), we have shown that, if m0(x) = o(x−3) as
x → +∞, then the upstream boundary condition f(ξ, τ) → 0 as τ → +∞.
However, we have not yet rigorously proved that the asymptotic steady solu-
tions are stable despite all the efforts we have made here. In Section 5.4, we
have tried to find numerically the best fit profiles of the solutions of (5.1),
which shows that the true profiles seem wandering within the family of steady
solutions. We have listed some reasons why we have not yet completely solved
this problem. After that, we have tried to apply the same method to the CH
93
CHAPTER 5. ASYMPTOTICS 5.6. CONCLUSIONS
equation in Section 5.5, which suggests that a more involved calculation will
be needed.
94
Chapter 6
Four Particle Systems
In this chapter, we will study the mCH in a very specific form. That is, we
will consider the Hamiltonian ODE system



q˙i = h
4∑
j=1
G(qi − qj)pj
p˙i = −hpi
4∑
j=1
G′(qi − qj)pj
(6.1)
where h = pi/2, G(x) is the Green function corresponding to the H2 metric
on S1 = [0, 2pi].
6.1 Motivation
Why do we study the four particle system? The answer is that we know that
the KdV equation and the CH equation are completely integrable but the
general mCH equation with k > 1 is very likely not, at the same time we
have two conserved quantities:
∫
m and
∫
mu for the general mCH equation
and an extra
∫
|m| 12 for the limiting mCH, so it is natural to ask if there
is another conserved quantity? The possible approaches to the question
include (i) constructing a conserved quantity explicitly and/or (ii) studying
the Lyapunov exponents of the corresponding four particle system.
For the particle systems, the conserved quantity
∫
|m| 12 dx becomes zero
(because the particle systems correspond to the evolution of the sum of some
δ momentum). If the four particle system has only one positive Lyapunov
exponent, then it is expected that the systems have some other integral than∫
m and
∫
mu in general. We will use the Lyapunov exponents method
95
CHAPTER 6. FOUR PARTICLE SYSTEMS 6.2. LYAPUNOV EXPONENTS
to show that this ODE system is likely to have another conserved quantity
than the obvious conserved quantities corresponding to
∫
m and
∫
mu. At the
same time, a positive Lyapunov exponent means the four particle system is
not integrable and hence the mCH equation is very unlikely to be integrable.
6.2 Lyapunov Exponents
Lyapunov exponents (or characteristic numbers) were first introduced by
Lyapunov [73] in 1892 to study the stability of nonstationary solutions of
ODEs and many papers and books (see, for example Nemyskii et al. [91], E.
Ott [93] and the reference therein) are devoted to it.
Let X be a differentiable manifold with a Riemannian metric, then an
ODE x˙ = f(t, x) on X defines a (flow) mapping F (t, ·) : X 7→ X for any
t > 0. The Lyapunov exponents are introduced to describe the long-time
dependence of the solution on the initial perturbation. More precisely, if
V ∈ Tx0M for some x0 ∈ M, dF (t, x) stands for the differential of F w.r.t.
x, then the Lyapunov is defined by
λ(x0, V ) ≡ limt→∞
1
t log ||dF (t, x0)(V )||. (6.2)
6.3 How to Compute Them?
Specifically to an ODE in Rn,
dx
dt = f(t, x), x(0) = x0 ∈ R
n, (6.3)
its linearised equation reads
dy(t)
dt = A(t)y(t), y(t) ∈ R
n, (6.4)
whose fundamental solution matrix Y (t) satisfies
dY (t)
dt = A(t)Y (t), Y (0) = Y0 ∈ R
n×n is orthogonal. (6.5)
If {pi} is an orthonormal basis of Rn, then
λi = lim sup
t→∞
1
t log(||Y (t)pi||) i = 1, 2, · · · , n, (6.6)
96
CHAPTER 6. FOUR PARTICLE SYSTEMS 6.3. HOW TO COMPUTE THEM?
are well-defined. When the sum
∑n
i=1 λi is minimised, the orthonormal basis
{pi} is called normal and the λi are the so-called Lyapunov exponents.
It is clear from above that the concept of Lyapunov exponents is a sort of
generalisation of the (real part of ) the eigenvalues for the constant coefficient
matrix A(t) ≡ A in the asymptotic stability analysis of y˙ = A(t)y, so one can
expect it will play a very important role in the study of asymptotic behaviour
of ODEs. That is why so many existing literatures are devoted to it.
Then how to compute them?
It is easy to get that for any normal basis we have
n∑
i=1
λi ≥ lim sup
t→∞
1
t
∫ t
0
trace(A(s))ds = lim sup
t→∞
1
t log |detY (t)|. (6.7)
If this inequality becomes an equality for some normal basis, then the lin-
ear system is called regular. One can find that the Lyapunov exponents
are invariant if we change Y (t) to another matrix X(t) = T (t)Y (t) with
T (t), T−1(t) are uniformly bounded. Perron and Diliberto (see [30]) show
that for bounded continuous A(t), there is an orthogonal Q(t) such that
X(t) = QT (t)Y (t) satisfies
dX(t)
dt = A˜(t)X(t), (6.8)
where A˜(t) is an upper triangular matrix. The reason we transform the
coefficient matrix A(t) to an upper triangular matrix A˜(t) becomes clear
when we look at the following theorem[73]:
Theorem 6.1 If A(t) ∈ Rn×n(t) is an upper triangular matrix with all en-
tries continuous and bounded, then the equation (6.1) is regular if and only
if the following limits exist
µi = limt→∞
1
t
∫ t
0
Aii(s)ds, i = 1, 2, · · · , n, (6.9)
and in this case, λi = µi, i = 1, 2, · · · , n.
The key point in computing Lyapunov exponents is the continuous QR de-
composition of Y (t),
Y (t) = Q(t)R(t), (6.10)
where Q(t) is orthogonal and R(t) is upper triangular with positive diagonal
entries Rii, i = 1, 2, · · · , n. From the orthogonality of Q(t), we have
λi = limt→∞
1
t log ||Y (t)pi|| = limt→∞
1
t log ||R(t)pi||. (6.11)
97
CHAPTER 6. FOUR PARTICLE SYSTEMS 6.4. FOUR PARTICLE SYSTEMS
So by Theorem 6.1,
λi = limt→∞
1
t log |Rii(t)|, 1 ≥ i ≥ n. (6.12)
From this, G. Benettini et al. [11] proposed the now very popular discrete
QR decomposition method in computing the Lyapunov exponents (L. Dieci
et al. [31] proved the convergence of this algorithm):
Discrete QR method: The point here is to QR decompose Y (t) indirectly
at t0 < t1 < · · · < tj < · · · . More precisely, let
Y0 ≡ Q0 = I,
and for j = 0, 1, · · · , one solves
Z˙j = AZj , Zj(tj) = Qj , tj ≤ t ≤ tj+1,
and then QR factorise Zj(tj+1)
Zj(tj+1) = Qj+1Rj+1,
with Rj+1 having positive diagonal entries. Since Q0 = I and
Y˙ = AY, Y (0) = I,
so if we denote Yj = Y (tj),, then we have Y˙ = AY, Y (tj) = Yj and
Yj+1 = Zj(tj+1)QTj Yj = Qj+1Rj+1QTj Yj = · · ·
= Qj+1Rj+1Rj · · ·R1Q0 = Qj+1
1∏
k=j+1
Rk.
The Lyapunov exponents can thus be obtained by
λi = limj→∞
1
tj
log ||(Rj)ii · · · (R1)ii|| = limj→∞
1
tj
j∑
k=1
log ||(Rk)ii||. (6.13)
6.4 Four Particle Systems
In order to study the Lyapunov exponents of (6.1) by the method of Dieci et
al. [31], we need to linearise the equation (6.1). The linearised equations are
98
CHAPTER 6. FOUR PARTICLE SYSTEMS 6.4. FOUR PARTICLE SYSTEMS
dY
dt = JY, (6.14)
where J is the matrix
J = pi2
[
A B
C D
]
, (6.15)
and
A =


∑
j 6=1
G′(q1 − qj)pj −G′(q1 − q2)p2 −G′(q1 − q3)p3 −G′(q1 − q4)p4
G′(q1 − q2)p1
∑
j 6=2
G′(q2 − qj)pj −G′(q2 − q3)p3 −G′(q2 − q4)p4
G′(q1 − q3)p1 G′(q2 − q3)p2
∑
j 6=3
G′(q3 − qj)pj −G′(q3 − q4)p4
G′(q1 − q4)p1 G′(q2 − q4)p2 G′(q3 − q4)p3
∑
j 6=4
G′(q4 − qj)pj


= −DT (6.16)
B =


G(0) G(q1 − q2) G(q1 − q3) G(q1 − q4)
G(q1 − q2) G(0) G(q2 − q3) G(q2 − q4)
G(q1 − q3) G(q2 − q3) G(0) G(q3 − q4)
G(q1 − q4) G(q2 − q4) G(q3 − q4) G(0)


(6.17)
C =


−p1
∑
j 6=1
G′′(q1 − qj)pj p1p2G′′(q1 − q2) p1p3G′′(q1 − q3) p1p4G′′(q1 − q4)
p1p2G′′(q1 − q2) −p2
∑
j 6=2
G′′(q2 − qj)pj p2p3G′′(q2 − q3) p2p4G′′(q2 − q4)
p1p3G′′(q1 − q3) p2p3G′′(q2 − q3) −p3
∑
j 6=3
G′′(q3 − qj)pj p3p4G′′(q3 − q4)
p1p4G′′(q1 − q4) p2p4G′′(q2 − q4) p3p4G′′(q3 − q4) −p4
∑
j 6=4
G′′(q4 − qj)pj


(6.18)
We can form the matrix J as follows: first construct the Hessian matrix
H ′′ of the Hamiltonian functional
H = 12
4∑
i,j=1
pipjG(qi − qj), (6.19)
and then get J by
J =
[
O I
−I O
]
×H ′′. (6.20)
We try different initial momentum values at four equi-spaced points (ie,
with qk(0) = kpi2 for k = 0, 1, 2, 3). The numerical results are shown in the
following figures. The bottom plot is a zoom-in of the top one in each figure.
99
CHAPTER 6. FOUR PARTICLE SYSTEMS 6.5. CONCLUSIONS
T=1500 in all the simulations. There do exist some cases for which the
Lyapunov exponents are convergent to zeros, but some other figures indicate
that the situations seem different. The Lyapunov exponent is notorious at its
convergence rate when computed, however, from these figures, the following
assertion is quite certain: at most one of the Lyapunov exponents is positive
as the time t→∞ !
The simulation suggests that although the four particle systems are not
integrable, there should exist another conserved quantity, but we have not
yet found any good candidate.
6.5 Conclusions
We have considered a four particle system corresponding to the limiting mCH
on S with m = (1 − ∂2x + ∂4x)u. We have used the method from [31] to
compute the Lyapunov exponents of the system of four particles initially
equi-distributed on the circle S. The numerical result suggests that the four
particle system is very likely non-integrable and at the same time it seems
to have another conserved quantity other than
∫
m and
∫
mu. A possible
candidate for the third integral is
∫
u(u2 + u2x + u2xx) (because we know that∫
u(u2 + u2x) is the third integral of the CH equation), and actually I once
believed I found a “proof” of that, but the numerical check disconfirmed that
and then I found that the “proof” was wrong!
100
CHAPTER 6. FOUR PARTICLE SYSTEMS 6.5. CONCLUSIONS
0 2 4 6 8 10
−15
−10
−5
0
5
10
15
Ly
ap
un
ov
 ex
po
ne
nts
1/t
0 0.005 0.01 0.015 0.02
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Ly
ap
un
ov
 ex
po
ne
nts
1/t
Figure 6.4.1: Lyapunov exponents for p = [1, 5, 10, 4]′: one has a positive
limit, six exponents go to zero as t→∞.
101
CHAPTER 6. FOUR PARTICLE SYSTEMS 6.5. CONCLUSIONS
0 2 4 6 8 10
−15
−10
−5
0
5
10
15
Ly
ap
un
ov
 ex
po
ne
nts
1/t
0 0.005 0.01 0.015 0.02
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Ly
ap
un
ov
 ex
po
ne
nts
1/t
Figure 6.4.2: Lyapunov exponents for p = [1, 5, 1, 10]′: one has a positive
limit, six exponents go to zero
102
CHAPTER 6. FOUR PARTICLE SYSTEMS 6.5. CONCLUSIONS
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
−10
−5
0
5
10
1/t
Ly
ap
un
ov
 ex
po
ne
nts
0 0.005 0.01 0.015 0.02
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
1/t
Ly
ap
un
ov
 ex
po
ne
nts
Figure 6.4.3: Lyapunov exponents for p = [2, 10, 0, 5]′: p3(0) = 0 implies
p3(t) ≡ 0 giving another conserved quantity. All exponents go to 0.
103
CHAPTER 6. FOUR PARTICLE SYSTEMS 6.5. CONCLUSIONS
0 1 2 3 4 5 6 7 8 9 10
−5
−4
−3
−2
−1
0
1
2
3
4
5
Ly
ap
un
ov
 ex
po
ne
nts
1/t
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
Ly
ap
un
ov
 ex
po
ne
nts
1/t
Figure 6.4.4: Lyapunov exponents for p = [1, 2, 3, 4]′: lower H||p||2 means
more likely to have quasiperiodic motion (ie, λ = 0).
104
Chapter 7
Higher Dimensional Case
7.1 Introduction
Up to now, all we have studied is about the one dimensional generalised Eu-
ler equations. But as we mentioned in section 1.5 of Chapter 1, the higher
dimensional generalised Euler equations can find many applications in Com-
putational Anatomy [51, 53]. So we turn to the two dimensional case.
In this chapter we will study the so-called two dimensional Camassa-Holm
equation, i.e., the Euler equation for G = Diff(R2) with H1(Rn) metric
∂
∂tm + u · ∇m +∇u
T ·m + m(divu) = 0. (7.1)
It was derived in [52] as a higher dimensional generalization of the 1d Camassa-
Holm equation, where the momentum density m(x, t) = (I −∆)u(x, t) is a
function from R2 × R 7→ R2. Holm and Marsden [51] mentioned its well-
posedness in Hs(R2) can be established by extending the arguments in [34],
which are mainly the geometric approach, and will be given in another publi-
cation (but I have not yet found any). Here we will prove the local existence
of classical solutions by an analysis approach.
7.2 Local Well-posedness
We will use some regularization method to establish the well-posedness.
First, we will give some properties of the mollifiers. Given any radial function
ρ(|x|) ∈ C∞0 (R2), ρ ≥ 0,
∫
R2
ρ = 1,
105
CHAPTER 7. HIGHER DIMENSIONAL CASE 7.2. LOCAL WELL-POSEDNESS
and define the mollification
(Jεv)(x) = ε−2
∫
R2
ρ(x− yε )v(y)dy, ε > 0.
Lemma 7.1 ([74]) The mollifier Jε defined above has the properties:
(i) ∀ u ∈ Lp(R2), 1 < p <∞, Jεu is a C∞ function.
(ii) For all v ∈ C0(R2), we have Jεv → v uniformly on any compact set Ω
in R2 and
|Jεv|L∞ ≤ |v|L∞.
(iii) DαJεv = JεDαv, ∀ |α| ≤ m, v ∈ Hm(R2).
(iv) For all u ∈ Lp(R2), v ∈ Lq(R2), 1p + 1q = 1, we have
∫
R2
(Jεu)vdx =
∫
R2
(Jεv)udx.
(v) For all v ∈ Hs(R2), Jεv converges to v in Hs and the rate of the con-
vergence in the Hs−1 norm is linear in ε,
lim
ε→0
||Jεv − v||s = 0,
||Jεv − v||s−1 ≤ Cε||v||s.
(vi) For all v ∈ Hm(R2), k ∈ Z+ ∪ {0}, and ε > 0, we have
||Jεv||m+k ≤
cmk
εk ||v||m,
|JεDkv|L∞ ≤
ck
ε1+k ||v||0.
Now we approximate the 2d CH equation (7.1) by the following smoothed
equations
∂
∂tm
ε +Jε(Jεuε ·∇Jεmε) +Jε((∇Jεuε)T · Jεmε) +Jε(Jεmε(divJεuε)) = 0,
(7.2)
mε = (I −4)uε, (7.3)
with mε(0) = m0(x) given.
106
CHAPTER 7. HIGHER DIMENSIONAL CASE 7.2. LOCAL WELL-POSEDNESS
Let
F (mε) = −Jε(Jεuε · ∇Jεmε)− Jε((∇Jεuε)T · Jεmε)− Jε(Jεmε(divJεuε))
≡ F1(mε) + F2(mε) + F3(mε).
(7.4)
Then for any m ∈ Z+ ∪ {0}, we have
||F1(mε1)− F1(mε2)||m
= ||Jε(Jεuε1 · ∇Jεmε1)−Jε(Jεuε2 · ∇Jεmε2)||m
≤ ||Jε(Jεuε1 · ∇Jε(mε1 −mε2))||m + ||Jε(Jε(uε1 − uε2) · ∇Jεmε2)||m
≡ G1 +G2,
and
G1 ≤ |Jεuε1|L∞||Dm+1Jε(mε1 −mε2)||0 + ||DmJεuε1||0|∇Jε(mε1 −mε2)|L∞
≤ 1ε |Jεuε1|L∞||mε1 −mε2||m + Cε2 ||DmJεuε1||0||m1 −m2||0
≤ Cε2 ||uε1||0||mε1 −mε2||m + Cε2 ||uε1||m||m1 −m2||0
≤ C(||m
ε
1||m)
ε2 ||mε1 −mε2||m.
Similarly, we have
G2 ≤
C
ε2 ||m
ε
1 −mε2||m. (7.5)
So we have
||F1(mε1)− F1(mε2)||m ≤
C
ε2 ||m
ε
1 −mε2||m, (7.6)
where C depends on ||mε1||m and ||mε2||m.
Analogously, we have
||Fi(mε1)− Fi(mε2)||m ≤
C
ε2 ||m
ε
1 −mε2||m, i = 2, 3. (7.7)
So, if we define an open subset O of Hm(R2) by O ≡ {m ∈ Hm(R2) :
||m||m < M } for some fix M > 0, then from the Picard Theorem, there
107
CHAPTER 7. HIGHER DIMENSIONAL CASE 7.2. LOCAL WELL-POSEDNESS
exists a T = Tε > 0 depending only on M and ε such that the smoothed
equations (7.2) admit a unique solution in C1((−T, T );O), ie a unique solu-
tion in C1((−T, T );Hm(R2)).
We take the derivative Dα, |α| ≤ m, m ≥ 2 of the equation (7.2) and
then inner product with Dαmε, use the fact
〈Jεuε · ∇DαJεmε, DαJεmε〉0 = −
1
2〈divJεu
εDαJεmε, DαJεmε〉0, (7.8)
we have (omitting the superscript ε and Jε)
1
2
d
dt ||D
αm||2L2 = −
〈
Dα(u∇m) +Dα(∇uT ·m) +Dα(mdivu), Dαm
〉
0 ,
(7.9)
We estimate the right side terms one by one.
〈Dα(u∇m), Dαm〉0 = 〈∇mDαu + u∇Dαm + other terms Q,Dαm〉0 ,
where Q is some intermediate terms between ∇mDαu and u∇Dαm. We can
estimate
〈∇mDαu, Dαm〉0 ≤ |Dαu|L∞||m||2m, (7.10)
and from (7.8) we have
|〈u∇Dαm, Dαm〉0| ≤ |divu|L∞||m||2m. (7.11)
Similarly we can estimate 〈Q,Dαm〉. The point here is that we need to
cancel the terms involving the m+ 1-th order derivative of m. If there is no
such higher order terms, then the estimate is straightforward. That is the
estimate for the first term in (7.9). We can obtain the similar estimates for
the other two terms in (7.9). Summing the resulting inequalities, we have
d
dt ||m||2m ≤ C(m)
∑
|α|≤m
〈Dαu∇m +Dα∇u ·m+
∇uDαm +Dαmdivu + mdivDαu, Dαm〉0
(7.12)
d
dt ||m||
2
m ≤ C(m)(|∇u|L∞||m||2m + |Dmu|L∞||m||2m + |m|L∞||u||m+1||m||m).
(7.13)
So Sobolev embedding theorems give us
d
dt ||m||m ≤ C||m||
2
m (7.14)
108
CHAPTER 7. HIGHER DIMENSIONAL CASE 7.2. LOCAL WELL-POSEDNESS
and integrating yields
||m(·, t)||m ≤
||m0||m
1− Ct||m0||m
, (7.15)
which means that for 0 < t < T ∗ = 1C||m0||m , ||m
ε(·, t)||m has an upper
bound independent of ε.
For any ε > 0, the equation (7.2) has a unique solution on [0, Tε), and
by the Continuation of an Autonomous ODE on a Banach Space ([74], page
103) we can extend the existence time interval to [0, T˜ε) such that T˜ε = ∞
or
lim
t→T˜ε
||mε(·, t)||m =∞.
On the other hand, from the inequality (7.15), we find that for 0 < t <
T ∗, ||mε(·, t)||m has an upper bound independent of ε. So we have
[0, T ∗) ⊂
⋂
ε>0
[0, T˜ε). (7.16)
Proposition 7.2 If {mε(x, t)} is a bounded set in Hm(R2) for all t ∈ [0, T ]
and some m ≥ 2, then there exist a constant C = C(||m0||m, T ) and a
constant 0 < δ < 1 such that for all ε, ε′ > 0, we have
sup
0≤t<T
||mε −mε′ ||0 ≤ C max(ε, ε′)1−δ. (7.17)
Proof From the equation (7.2) we have
1
2
d
dt ||m
ε −mε′||20 = 〈F (mε)− F (mε
′),mε −mε′〉0
=
3∑
i=1
〈Fi(mε)− Fi(mε
′),mε −mε′〉0.
(7.18)
Let us take the second term as an example, and the other two terms are
proved analogously.
〈F2(mε)−F2(mε
′),mε−mε′〉0 = −〈(∇Jεuε)T ·Jεmε−(∇Jε′uε
′)T ·Jε′mε
′ ,mε−mε′〉0
= −
〈
(∇Jεuε)T · Jε(mε −mε
′) + (∇Jεuε)T · (Jε − Jε′)mε
′+
(∇Jε(uε − uε
′)T · Jε′mε
′ + (∇(Jε − Jε′)uε
′)T · Jε′mε
′, mε −mε′
〉
0
= T1 + T2 + T3 + T4.
109
CHAPTER 7. HIGHER DIMENSIONAL CASE 7.2. LOCAL WELL-POSEDNESS
|T1| = 〈(∇Jεuε)T · Jε(mε −mε
′), mε −mε′〉0
≤ C||mε −mε′ ||20||∇uε||L∞ ≤ C||mε −mε
′ ||20||mε||2.
Similarly,
|T3| ≤ C||mε −mε
′ ||0||mε −mε
′||2||mε
′||0.
From Lemma 7.1, we have
|T2| ≤ C max(ε, ε′)||mε −mε
′||0||mε||2||mε
′||1,
For some 0 < δ < 1, we have
|T4| ≤ C||mε −mε′ ||0||mε′||0||∇(Jε − Jε′)uε′ ||L∞
≤ C||mε −mε′ ||0||mε′||0||∇(Jε − Jε′)uε′ ||H1+δ
≤ C||mε −mε′ ||0||mε
′||0||(Jε − Jε′)mε
′ ||H1+δ
≤ C max(ε, ε′)1−δ||mε −mε′||0||mε
′||0||mε
′||2,
here we have used the following inequality which can be easily proved:
||Jεv − v||s−δ ≤ Cεδ||v||Hs, for all v ∈ Hs(R2).
So if ||mε(x, t)||m ≤ M for some constant M > 0 and for all 0 < t < T, then
the above inequalities yield
d
dt ||m
ε −mε′ ||0 ≤ C(max(ε, ε′) + max(ε, ε′)1−δ + ||mε −mε
′ ||0), (7.19)
and Gronwall inequality tells us
sup
0≤t≤T
||mε −mε′||0 ≤ C max(ε, ε′)1−δ (7.20)
because mε have the same initial value.
We have just proved that the existence of an m such that
sup
0≤t≤T
||mε −m||0 ≤ Cε1−δ. (7.21)
Moreover, from the Interpolation of Sobolev Spaces:
||v||s′ ≤ Cs||v||1−s
′/s
0 ||v||s
′/s
s , for 0 < s′ < s,
110
CHAPTER 7. HIGHER DIMENSIONAL CASE 7.2. LOCAL WELL-POSEDNESS
we have for 0 < r < m
sup
0≤t≤T
||mε −m||r ≤ C(||m0||, T )ε(1−δ)(1−r/m) (7.22)
Hence we have the strong convergence in C([0, T ];Hr(R2)). If 0 < 2 < r < m,
this implies strong convergence in C([0, T ];C1(R2)). In the equation (7.2),
the last three terms of (7.2) converge in C(0, T ;C(R2)), so does the first
term of it, ie, mεt , which means that,if m0 ∈ Hm(R2) with m > 2, then the
equation (7.1) has a unique classical solution for 0 < t < T.
111
Chapter 8
Future Work
´ûûÙ?ñ§Æòþe¦¢"
——¯§5lÖ6§úc290c"
The way ahead is long; I see no ending, yet high and low, I’ll search
with my will unbending.
—Qu Yuan, Li Sao, ∼ 290 BC.
8.1 Stability of the Asymptotic Solutions
Our numerical simulation shows that the solution of the equation
mt + umx + 2uxm = 0, (8.1)
where (1− ∂2x)ku = m, k = 2, with the Gaussian initial values, tends to the
asymptotic solution
f(a, η) = C(a)e
−2a[arctan(a(η−1))+pi2 ]
((η − 1)2 + a−2)2 , (8.2)
with some constant C(a), and
lim
a→+∞
f(a, η) = f˜ =



4(η − 1)−4e 2η−1 if η − 1 < 0,
0, if η − 1 > 0.
(8.3)
But we have not yet found the rigorous explanation of this phenomena. In
Chapter 5, we tried to use the characteristics method to understand how this
weak blowup forms, but have not solved this completely.
112
CHAPTER 8. FUTURE WORK 8.1. STABILITY OF THE ASYMPTOTIC SOLUTIONS
There are lots of literature devoted to the stability of steady solutions of
some evolutionary equations. For example, the blowup rate and profile for
the nonlinear heat equation
ut = 4u+ up in RN
with some restrictions on p, has been determined by Giga and Kohn [43]
and Fermanian-Kammerer et al. [38]. However, the existence of Lyapunov
functions plays a fundamental role in their stability study of the blowup
profile. Another approach is the so-called Evans function method introduced
by J. Evans [36], whose application in stability of travelling wave solutions
of the dissipative systems is summarised by T. Kapitula [60].
The Evans function method also finds some successful applications to
Hamiltonian PDEs. R. Pego and M. Weinstein [94] used the Evans function
to discuss the instability of solitary waves. There are many other approaches
as well to the stability problem for the Hamiltonian PDEs. For example,
Grillakis et al. [48] established some sharp conditions for the stability and
instability of solitary waves of some Hamiltonian PDEs. A. Constantin and
W. Strauss [26], J. Lenells [69] exploited the first three conserved quantities to
establish the stability of peakons for Camassa-Holm equations on the whole
R1 and periodic case respectively. Another important contribution to the
stability of the blowup profile is made by Y. Martel and F. Merle [80]– [83],
where they introduced some new tools to study the stability of the blowup
profile of the generalised KdV equation
ut + (uxx + up)x = 0, (t, x) ∈ R+ ×R.
But all the approaches above cannot directly apply to our case. The
equation (8.1) is indeed a Hamiltonian PDE, but the functions in (8.2) and
(8.3) do not correspond to the travelling wave solutions of (8.1), ie, tf(t(x−
ct)) is not a solution of (8.1), it is only an asymptotic solution! If we consider
the asymptotic slow time PDE (5.11):
fτ + (ξf)ξ + Ufξ + 2Uξf = 0, (8.4)
here U = G′′(0)2 (ξ2f0−2ξf1 +f2). Then these f ’s are steady solutions of (8.4),
but now the PDE (8.4) is no longer a Hamiltonian PDE, and the fact that
the coefficients in (8.4) depend on the integrals of f makes things even more
difficult.
113
CHAPTER 8. FUTURE WORK 8.2. POSITIVITY OF SOLUTIONS
8.2 Positivity of Solutions
For the periodic Camassa-Holm equation, it is well-known that the positivity
of the initial momentum implies the global well-posedness and the positivity
of the momentum for any time t > 0. The proof depends on the complete
integrability [25] [65]. Our numerical simulation supports that the equation
(8.1) has this property as well, which is consistent with the geometric ob-
servation that the generalised Euler equation preserves the co-adjoint orbits,
whereas we do not know whether it is an integrable system or not. This
means that the preservation of the positivity of momentum may not relate
to the complete integrability.
8.3 Higher Dimensional Case
In Chapter 7 we discussed the local well-posedness of the Camassa-Holm
equation, but we do not know anything about the global well-posedness and
other properties. For the torus T2 case, we conjecture that if all components
of the initial momentum have no simultaneous zero point (ie, the momentum
vector is never zero), then the solution should exist globally. But we have no
idea how to prove this.
Another question is to study the dynamics of the higher-dimensional gen-
eralised Euler equations and how they depend on the choice of the metric.
114
Appendix A
Properties of the Green’s
Function
Lemma A.1 Let
G(x) = 1 + 2
∞∑
n=1
1
1 + n2 + n4 cos(nx) x ∈ S = [0, 2pi],
then ∂3xG(x) ∈ L∞(S).
Proof We know from Abel-Dirichlet criterion [107] that
(or see http://en.wikibooks.org/wiki/Real Analysis/Series)
∞∑
n=1
n3
1 + n2 + n4 sin(nx)
converges for any x ∈ S, and uniformly converges in any [α, β] ⊂ (0, 2pi) if
0 < α < β < 2pi. That means the Fourier series
2
∞∑
n=1
n3
1 + n2 + n4 sin(nx)
converges to ∂3xG :
∂3xG(x) = 2
∞∑
n=1
n3
1 + n2 + n4 sin(nx). (A.1)
On the other hand, we know
∞∑
n=1
sin(nx)
n =
pi
2 (1−
x
pi ) for 0 < x < 2pi, (A.2)
115
APPENDIX A. PROPERTIES OF THE GREEN’S FUNCTION
and if we denote g(x) for this function, then
|g(x)− ∂3xG(x)| =
∣∣∣∣∣
∞∑
n=1
( 1
n −
n3
1 + n2 + n4
)
sin(nx)
∣∣∣∣∣ (A.3)
which converges uniformly to a bounded function on S. So we have
∂3xG ∈ L∞(S).
Now we are going to discuss some properties of the Green’s function of
the operator Λ44 acting on H∞(R1), that is,
Λ44G(x) = (I − ∂2x)2G(x) = δ on R1. (A.4)
The Green’s function G(x) can be expressed (up to a multiplicative constant)
via Fourier transform
G(x) =
∫
R1
1
1 + 2|ξ|2 + |ξ|4 exp(ixξ)dξ
=
∫
R1
1
1 + 2|ξ|2 + |ξ|4 cos(xξ)dξ.
(A.5)
Clearly, G(x) is uniformly continuous on R1 and G ∈ C2+ε(R1) for any
0 ≤ ε < 1.
Lemma A.2 Let G(x) be the Green’s function as above, then
||∂3xG||L∞(R1) <∞. (A.6)
Proof The idea is similar to that in the proof of Lemma A.1.
G(x) =
∫
R1
1
1 + 2|ξ|2 + |ξ|4 cos(xξ)dξ, (A.7)
Abel-Dirichlet criterion for the improper integral tells us that
∫
R1
ξ3
1 + 2|ξ|2 + |ξ|4 sin(xξ)dξ
converges on R1 and uniformly converges on any bounded interval away from
x = 0, which means
∂3xG(x) = −
∫
R1
ξ3
1 + 2|ξ|2 + |ξ|4 sin(xξ)dξ. (A.8)
116
APPENDIX A. PROPERTIES OF THE GREEN’S FUNCTION
Now let us introduce a bounded function
g(x) =



ex, x < 0
−e−x, x > 0
(A.9)
then it is easy to find the Fourier transform of g is 2i ξ1+|ξ|2 , which means
g(x) =
∫
R1
2i ξ1 + ξ2 exp ixξdξ
= −2
∫
R1
ξ
1 + ξ2 sin xξdξ.
(A.10)
On the other hand,
∣∣∣∣∂3xG(x)−
1
2g(x)
∣∣∣∣ =
∣∣∣∣
∫
R1
( ξ
1 + ξ2 −
ξ3
1 + 2|ξ|2 + |ξ|4
)
sin xξdξ
∣∣∣∣ (A.11)
is clearly convergent to a C1 function on R1 and so is bounded on any bounded
interval. Moreover, the Riemann-Lebesgue Lemma tells us that
lim
|x|→∞
∣∣∣∣∂3xG(x)−
1
2g(x)
∣∣∣∣ = 0, (A.12)
so
|∂3xG|L∞(R1) <∞.
Lemma A.3 The Green’s function of Λ = (1− ∂2x)2 on R1 is
G(x) = 14(|x|e
−|x| + e−|x|), (A.13)
so it is positive on R1.
Proof We use the method of undetermined coefficients to solve the equation
(1− ∂2x)2G(x) = δ.
First, find the eigenvalues of this operator, λ = ±1 (double eigenvalues), so
the solutions have the form of h(x) = aex + bxex + ce−x + dxe−x. Then we
impose the requirements
h(0+) = h(0−); h′(0+) = 0 = h′(0−);
h′′(0+) = h′′(0−); h′′′(0−) = −12 = −h′′′(0+)
117
APPENDIX A. PROPERTIES OF THE GREEN’S FUNCTION
to find that the solution is G(x) = 14e−|x|(1 + |x|).
The arguments here apply to the Green’s function of the general initial
operator and we have
Lemma A.4 Let G(x) be the Green’s function of the operator Λ2k2k = (1−∂2x)k
with k ≥ 1, then G(x) ≥ 0 and
|∂2k−1x G|L∞ <∞. (A.14)
Proof The idea is same as before and we can find the Green function in
this case is
G(x) = 12ke
−|x|(1 + |x|+ · · ·+ |x|k−1).
118
Appendix B
Multi-symplectic Formulation
In this appendix, we will establish a multi-symplectic formulation for the
mCH equation via the multi-symplectic geometry. [45] provides the details
of the foundation of multi-symplectic geometry.
B.1 Multi-symplectic Geometry
We recall some aspects of the so-called multi-symplectic geometry. Let X be
an orientable n + 1 dimensional manifold (which in our case is S × R1) and
let piXY be a fiber bundle over X, which we call the covariant configuration
bundle. The space of sections of piXY will be denoted by C∞(piXY ) (which in
our case is the space of smooth functions on S × R1.)
Definition B.1 The first jet bundle J1(Y ) of Y is the affine bundle over
Y whose fiber over y ∈ Yx consists of those linear mappings γ : TxX 7→ TyY
satisfying
TpiXY ◦ γ = identity on TxX. (B.1)
The map γ corresponds to Txφ for a local section φ because we can easily
find that
piXY (φ(x)) = x ∀x ∈ X =⇒ TpiXY ◦ Txφ(x)(v) = v ∀v ∈ TxX. (B.2)
If X has local coordinates xν , ν = 1, 2, · · · , n, adapted coordinates on Y
are yA, A = 1, 2, · · · , N, along the fiber Yx. Then Coordinates (xν , yA) induce
coordinates yAν on the fibers of J1(Y ). The map x 7→ Txφ defines a section of
J1(Y ) regarded as a bundle over X, and we denote this section by j1(φ) and
call it the first jet of φ or the first prolongation of φ. In coordinates, j1(φ) is
given by
xν 7→ (xν , φA(xν), ∂µφA(xν)),
119
APPENDIX B. MULTI-SYMPLECTIC FORMULATION B.1. MULTI-SYMPLECTIC GEOMETRY
where ∂µ = ∂∂xµ . A section of the bundle J1(Y ) 7→ X which is the first jet
of a section of piXY is said to be holonomic. We can define the higher order
jet bundles by induction: J2(Y ) ≡ J1(J1(Y )), Jk(Y ) ≡ J1(Jk−1(Y )) for any
integer k ≥ 2. Introduce the k-th jet prolongation jk(φ) ≡ j1(jk−1(φ)) of a
section φ : X 7→ Y. In coordinates, j2(φ) is given by
xν 7→ (xν , φA(xν), ∂µφA(xν), ∂ν1∂ν2φA(xν)).
A section ρ of Jk(Y ) 7→ X is said to be k-holonomic if ρ = jk(piY,Jk(Y ) ◦ ρ).
If, for example, X = S × R1, Y = X × R1, φ ∈ C∞(piXY ) means that
φ = φ(x, t) is a smooth function, and j1(φ) means a map
(x, t) 7→ (x, t, φ(x, t), φx(x, t), φt(x, t)), (B.3)
and j2(φ) means a map
(x, t) 7→ (x, t, φ(x, t), φx(x, t), φt(x, t), φxx(x, t), φxt(x, t), φtx(x, t), φtt(x, t)).
(B.4)
We use jk(φ) for these maps and their values jk(φ)(x, t) if no confusions
occur.
Define the set
C∞ ≡
{
φ : X 7→ Y
∣∣ ∀x ∈ X, there is a smooth open manifold U ⊂ X with
smooth closed boundary, such that piXY ◦ φ|U : U 7→ X is an embedding} .
For each φ ∈ C∞ set φX ≡ piXY ◦φ and UX ≡ φX(U) so that φX : U 7→ UX
is a diffeomorphism. Let C be the closure of C∞ in some Hilbert norm. Denote
Ck ≡ {jk(φ ◦ φ−1X )| φ ∈ C}.
The tangent space to the manifold C at a point φ ∈ C is
{
V ∈ C∞(X, TY )| locally piY,TY ◦ V = φ and VX ≡ TpiXY ◦ V ◦ φ−1X
is a vector field on UX} ,
which means that it consists of vector fields V such that the diagram in the
Figure B.1.1 commutes.
We introduce
G ≡ {ηY : Y 7→ Y | ηY covers a diffeomorphism ηX : X 7→ X,
ηY is a piXY bundle automorphism} .
120
APPENDIX B. MULTI-SYMPLECTIC FORMULATION B.1. MULTI-SYMPLECTIC GEOMETRY
Y TY TX
U(⊂X) UX
piY,TY TpiXY
φ
V VX
(φX)−1
Figure B.1.1: The tangent vector V ∈ TφC
TY TX
X X
Y X
Y
TpiXY
V VX
piY,TY
φ
piXY
piXY
ηY
ηX
φX
Figure B.1.2: Various spaces and mappings
121
APPENDIX B. MULTI-SYMPLECTIC FORMULATION B.1. MULTI-SYMPLECTIC GEOMETRY
We can summarise the various sets and mappings in the Figure B.1.2.
We need introduce the vertical sub-bundle V Y of Y : it is defined as a
sub-bundle whose fiber over y ∈ Yx ≡ pi−1XY (x) is
VyY ≡ {v ∈ TyY : TpiXY · v = 0}. (B.5)
For γ ∈ J1y (Y ), we have the splitting
TyY = image of γ ⊕ VyY. (B.6)
We will have some intuitive ideas about these concepts through the fol-
lowing example:
Example B.2 If X = R1, Y = X × R1 = R2, then a smooth section f of
Y 7→ X means a smooth function f : R1 7→ R1. In other words, we have a
smooth mapping
f : x ∈ X = R1 7→ (x, f(x)) ∈ Y = R2
whose composition with piXY is fX = piXY ◦f =identity on R1. If denote Γ the
graph of f : Γ = {(x, y) : y = f(x), x ∈ R1}, and that V = (V1, V2) ∈ TpR2
for some p = (x, f(x)) ∈ Γ, then the first prolongation γ = j1(f) of f is a
mapping
j1(f) : (x, V1) ∈ TxR1 7→ (x, f(x), V1, f ′(x)V1) ∈ TpR2,
or in other words,
γ · V1 = (V1, f ′(x)V2) = V1
∂
∂x + f
′(x)V1
∂
∂y ∈ TpR
2.
So
V v = V − γ · V1 = (V2 − f ′(x)V1)
∂
∂y ∈ TpR
2
is a vector in the vertical sub-bundle of Y.
If we consider a mapping φ ∈ C instead of the section f above, then
locally, we have
φ : x 7→ (φX(x), f(x)) ∈ R2
with a diffeomorphism φX and a smooth function f , which means we have
a smooth section f(φ−1X (x)) of Y 7→ UX . In other words, φ ◦ (φX)−1 : x 7→
(x, f ◦φ−1X (x)) is a smooth section of Y 7→ UX , and we have the similar results
for this section φ ◦ (φX)−1.
122
APPENDIX B. MULTI-SYMPLECTIC FORMULATION B.1. MULTI-SYMPLECTIC GEOMETRY
V=(V1,V2)
V v
V h
y=f(x)
Figure B.1.3: The vertical part V v of a vector V
Without loss of generality, from now on, we take an open bounded domain
U ⊂ X and consider UX = φX(U) only.
Definition B.3 The action functional S : C 7→ R is given by
S(φ) =
∫
UX
L(jk(φ ◦ φ−1X )), φ ∈ C (B.7)
for some Lagrangian density L : Jk(Y ) 7→ Λn+1(X), and φ ∈ C is called an
extremum of S if
d
dλ
∣∣∣∣
λ=0
S(ηλY ◦ φ) = 0
for all smooth paths λ 7→ ηλY in G, where ηλY covers a diffeomorphism ηλX and
η0Y = idY .
For any φλ ∈ C such that φ0 = φ, and dφλdλ |λ=0 = V, there is ηλY : Y 7→ Y
such that φλ = ηλY ◦ φ. We can see that ηλY ◦ (φ ◦ φ−1X ) ◦ (ηλX)−1 is a section of
Y 7→ X, and it maps UλX ≡ ηλX ◦ φX(U) to φλ(U).
We are going to introduce the prolongations of automorphisms ηY of Y
and of elements V ∈ TφC.
Definition B.4 The first prolongation j1(ηY ) : J1(Y ) 7→ J1(Y ) of an
automorphism ηY of Y 7→ X is defined by (as shown in Figure B.1.4 ):
j1(ηY )(γ) = TηY ◦ γ ◦ Tη−1X .
123
APPENDIX B. MULTI-SYMPLECTIC FORMULATION B.1. MULTI-SYMPLECTIC GEOMETRY
Y Y TY TY
X X TX TX
ηY
ηX
piXY piXY γ j1(ηY )(γ)
TηY
TηX
Figure B.1.4: The prolongation of ηY
For a vector V ∈ TφC, let ηλY be the flow of a vector field v on Y with
v ◦ φ = V. Then the first prolongation j1(V ) of V is a vector field on
J1(Y ) defined by
j1(V ) = ddλ
∣∣∣∣
λ=0
j1(ηλY ).
We can define the k-th prolongations of an automorphism ηY and
a vector V ∈ TφC for all k ≥ 1 by induction.
Definition B.5 The variational derivative of f : Jk(Y ) 7→ R is the
function on J2k(Y ) defined by
δf
δyA ≡
k∑
s=0
(−1)sDµ1Dµ2 · · ·Dµs
( ∂f
∂yAµ1···µs
)
, (B.8)
where Dµ(f) ≡ ∂f∂xµ +
∂f
∂yA yAµ + · · ·+
∂f
∂yAµ1···µk
yAµ1···µkµ.
Definition B.6 The first dual jet bundle J1(Y )∗ is the vector bundle over
Y whose fiber at y ∈ Yx is the set of affine maps from J1(Y )y to Λn+1(X)x,
the bundle of n+ 1-forms on X. A smooth section of J1(Y )∗ is therefore an
affine bundle map of J1(Y ) to Λn+1(X) covering piXY .
Fiber coordinates on J1(Y )∗ are (p, pνA), which correspond to the affine
map given in coordinates by
vAν 7→ (p+ vAν pνA)dn+1x, (B.9)
where dn+1x = dx1 ∧ dx2 · · · ∧ dxn ∧ dx0.
124
APPENDIX B. MULTI-SYMPLECTIC FORMULATION B.1. MULTI-SYMPLECTIC GEOMETRY
Analogous to the canonical one- and two-forms on a cotangent bundle,
there exist canonical (n+ 1)- and (n+ 2)-forms on the jet bundle J1(Y )∗. In
coordinates, with dnxν ≡ ∂ν a dn+1x standing for the interior product of ∂∂xν
and dn+1x, these forms are given by
θ = pνAdyA∧dnxν +pdn+1x, Ω = −dyA∧dpνA∧dnxν + dp∧dn+1x. (B.10)
Similarly, we can define the k-th dual jet bundle Jk(Y )∗ and the canon-
ical one- and two-forms on it.
A Lagrangian density L : Jk(Y ) 7→ Λn+1(X) is a fiber preserving map:
L(jk(φ)) = L(jk(φ))dx1 ∧ dx2 · · · ∧ dxn ∧ dx0. (B.11)
For k-th order Lagrangian field theory, the fundamental geometric structure
is the Cartan form θL, which is defined as the pullback of the canonical
n+ 1-form θ on Jk(Y )∗ by (FL)∗:
θL ≡ (FL)∗θ, (B.12)
where the fiber derivative FL : Jk(Y ) 7→ Jk(Y )∗, expressed intrinsically as
the first order vertical Taylor approximation to L, is defined by
FL(γ) · γ′ = L(γ) + ddε
∣∣∣∣
ε=0
L(γ + ε(γ′ − γ)), (B.13)
where γ, γ′ ∈ Jk(Y ).
Now denote ω = dn+1x, ων = ∂∂xν a ω.
Theorem B.7 Given a smooth Lagrangian density L : Jk(Y ) 7→ Λn+1(X),
there exist a unique Ψ ∈ Λn+2(J2k(Y )) given by
Ψ = δLδyAdy
A ∧ ω,
a unique map DELL ∈ C∞(C2k, T ∗C ⊗ Λn+1(X)) given by
DELL(φ) · V = j2k(φ ◦ φ−1X )∗
( δL
δyA iV (dy
A ∧ ω)
)
, (B.14)
and a unique n+ 1 form θL ∈ Λn+1(J2k−1(Y )) given by
θL =
k∑
s=0
pµ1µ2···µsA dyAµ1···µs−1 ∧ ωµs
= pω + pµAdyA ∧ ωµ + p
µ1µ2
A dyAµ1 ∧ ωµ2 + · · ·+ p
µ1µ2···µk
A dyAµ1···µk−1 ∧ ωµk
(B.15)
125
APPENDIX B. MULTI-SYMPLECTIC FORMULATION B.1. MULTI-SYMPLECTIC GEOMETRY
such that for any V ∈ TφC and any open subset UX of X with UX
⋂
∂X = ∅,
we have
dSφ · V =
∫
UX
DELL(φ) · V +
∫
∂UX
j2k−1(φ ◦ φ−1X )∗[j2k−1(V ) a θL], (B.16)
where iV or V a denotes the interior product with V , and
p = L− ∂L∂yAµ1
yAµ1 +Dµ2
( ∂L
∂yAµ1µ2
)
yAµ1 −Dµ2Dµ3
( ∂L
∂yAµ1µ2µ3
)
yAµ1 + · · ·
+(−1)kDµ2 · · ·Dµk
( ∂L
∂yAµ1···µk
)
yAµ1
− ∂L∂yµ1µ2
yAµ1µ2 +Dµ3
( ∂L
∂yAµ1µ2µ3
)
yAµ1µ2 + · · ·
+(−1)k−1Dµ3 · · ·Dµk
( ∂L
∂yAµ1···µk
)
yAµ1µ2 + · · · −
∂L
∂yAµ1···µk
yAµ1···µk ,
pµA =
∂L
∂yAµ
−Dµ2
( ∂L
∂yAµµ2
)
+Dµ1Dµ2
( ∂L
∂yAµµ1µ2
)
+ · · ·
+(−1)k−1Dµ2 · · ·Dµk
( ∂L
∂yAµµ2···µk
)
pµ1µ2A =
∂L
∂yAµ1µ2
−Dµ3
( ∂L
∂yAµ1µ2µ3
)
+Dµ3Dµ4
( ∂L
∂yAµ1µ2µ3µ4
)
+ · · ·
+(−1)kDµ3 · · ·Dµk
( ∂L
∂yAµ1µ2···µk
)
,
· · · · · ·
pµ1µ2···µkA =
∂L
∂yAµ1µ2···µk
.
Moreover, the θL agrees with the n+ 1 form introduced by (B.12), and ΩL =
dθL is the multi-symplectic form on J2k−1(Y ). Furthermore,
DELL(φ) · V = j2k−1(φ ◦ φ−1X )∗[j2k−1(V ) a ΩL] in UX
and the variational principle (B.14) gives, on the interior of the domain, the
Euler-Lagrange equation
∂L(jk(φ ◦ φ−1X ))
∂yA −
∂
∂xµ1
(∂L(jk(φ ◦ φ−1X ))
∂yAµ1
)
+ · · ·+
· · ·+ (−1)k ∂
k
∂xµ1 · · ·∂xµk
(
∂L(jk(φ ◦ φ−1X ))
∂yAµ1···∂µk
)
= 0.
(B.17)
126
APPENDIX B. MULTI-SYMPLECTIC FORMULATION B.1. MULTI-SYMPLECTIC GEOMETRY
Proof J. Marsden et al [76] and S. Kouranbaeva et al [66] have proved the
result for k = 1 and k = 2 respectively, and their ideas work for the general
k ≥ 1 case although the calculation is quite involved. Here we omit the
details but only mention the main ideas as follows:
For any φλ ∈ C such that φ0 = φ, and dφλdλ |λ=0 = V, there is ηλY : Y 7→ Y
such that φλ = ηλY ◦ φ. We can see that ηλY ◦ (φ ◦ φ−1X ) ◦ (ηλX)−1 is a section
of Y 7→ X, and it maps UλX ≡ ηλX ◦ φX(U) to φλ(U) and piXY ◦ φλ ◦ φ−1X =
ηλX , VX =
dηλX
dλ |λ=0.
So we have
dSφ · V =
d
dλ
∣∣∣∣
λ=0
∫
UλX
L(jk(φλ ◦ (φλX)−1))
=
∫
UX
d
dλ
∣∣∣∣
λ=0
L(jk(φλ ◦ (φλX)−1))+
∫
UX
d
dλ
∣∣∣∣
λ=0
(ηλX)∗L(jk(φ ◦ (φX)−1))
, I + II.
(B.18)
Then use the Cartan’s formula to evaluate the second part
II =
∫
UX
d
dλ
∣∣∣∣
λ=0
(ηλX)∗L(jk(φ ◦ (φX)−1))
=
∫
UX
£VXL(jk(φ ◦ (φX)−1))
=
∫
UX
£VX (Lω) =
∫
UX
d ıVX (Lω) + ıVXd(Lω)
=
∫
∂UX
L ıVXω =
∫
∂UX
LV µωµ.
The first part
I =
∫
UX
d
dλ
∣∣∣∣
λ=0
L(jk(φλ ◦ (φλX)−1))
=
∫
UX
[ ∂L
∂yA (j
k(φ ◦ φ−1X ))(V v)A +
∂L
∂yAµ1
(jk(φ ◦ φ−1X ))(V v)Aµ1+
· · ·+ ∂L∂yAµ1···µk
(jk(φ ◦ φ−1X ))(V v)Aµ1···µk
]
ω,
127
APPENDIX B. MULTI-SYMPLECTIC FORMULATION B.1. MULTI-SYMPLECTIC GEOMETRY
here we have used that ddλφλ
∣∣
λ=0 = V, φ
λ
X = piXY ◦ φλ = piXY ◦ (ηλY ◦ φ) =
ηλX ◦ φX , so (φλX)−1 = φ−1X ◦ (ηλX)−1 and so
d
dλ
∣∣∣∣
λ=0
φλ ◦ (φλX)−1 =
d
dλ
∣∣∣∣
λ=0
φλ ◦ (φX)−1 + Tφ ·
d
dλ
∣∣∣∣
λ=0
(φλX)−1
= V ◦ φ−1X + T (φ ◦ φ−1X )(−VX)
= V ◦ φ−1X − T (φ ◦ φ−1X )VX = V v.
Then use integration by parts to directly check that (B.18) gives the interior
integral and boundary integral parts in the Theorem.
We can summarise the analogy between the classical mechanics and field
theory in the following table:
Analogy between classical mechanics and field theory
Classical Mechanics k-th Field Theory
Configuration Space Q the fiber bundle Y 7→ X
Phase Space TQ k-th jet bundle Jk(Y )
1-form / (n+ 1)-form θL = (FL)∗θ θL = (FL)∗θ for a covariant
for an L : TQ 7→ R L : Jk(Y ) 7→ Λn+1(X).
2-form /(n+ 2)-form symplecity: ωL = dθL multi-symplecity: ΩL = dθL
We call the critical points of the action functional S the solutions of the
Euler-Lagrange equation, and introduce the notations:
Definition B.8
P ≡ {φ ∈ C| j2k−1(φ◦φ−1X )∗ ıWΩL = 0 for all vector fields W on J2k−1(Y )}
(B.19)
denotes the solution space of the Euler-Lagrange equation.
For any φ ∈ P, let
F ≡
{
V ∈ TφC| j2k−1(φ ◦ φ−1X )∗£j2k−1(V )[W a ΩL] = 0
for all vector fields W on J2k−1(Y )
} (B.20)
128
APPENDIX B. MULTI-SYMPLECTIC FORMULATION B.2. FORMULATION FOR MCH
be the set of the solutions of the first variation equations of the Euler-Lagrange
equations, where £V stands for the Lie derivative along V.
Then we have the following result whose proof is similar to that in [66].
Theorem B.9 (Multi-symplectic form formula) If φ ∈ P, then for
all V and W in F , we have
∫
∂UX
j2k−1(φ ◦ φ−1X )∗[j2k−1(V ) a j2k−1(W ) a ΩL] = 0. (B.21)
B.2 Formulation for mCH
We adopt the Lagrangian approach to mCH which leads naturally to the
multi-symplectic formulation. For the modified Camassa-Holm equation with
H2 metric:
mt + 2uym+ umy = 0 for y ∈ S, m = Λ44u. (B.22)
This equation is the generalised Euler equation for the reduced Lagrangian:
l(u) ≡ 12
∫
(u2 + u2y + u2yy)dy. (B.23)
On the other hand, we can express it in the Lagrangian variable η(t, x) which
is the solution of 


∂
∂tη(t, x) = u(t, η(t, x)),
η(0, x) = x.
(B.24)
Now let X = S×R, Y = X×R = S×R×R with coordinates (x1, x0) = (x, t)
for X and (x1, x0, y) = (x, t, y) for Y . A smooth section φ of Y 7→ X is a
mapping (x, t) 7→ (x, t, η(t, x)), where η(t, x) is the solution of (B.24). The
material or Lagrangian velocity ∂∂tη(t, x) is an element of Tφ(t,x)Y = T(t,x,y)Y,
where y = η(t, x).
With (B.24) and uy = ηtx/ηx, uyy = ηtxxηx−ηtxηxxη3x the Lagrangian repre-sentation for the action may be expressed as
S(φ) = 12
∫
S×[0,T ]
(ηxη2t + η2tx/ηx + (ηxηtxx − ηtxηxx)2/η5x)dxdt. (B.25)
129
APPENDIX B. MULTI-SYMPLECTIC FORMULATION B.2. FORMULATION FOR MCH
The third jet bundle J3(Y ) is a 17 dimensional manifold and three-holonomic
sections of J3(Y ) 7→ X have local coordinates
j3(φ) = (t, x, η, ηt, ηx, ηtt, ηtx, ηxt, ηxx, ηttt, ηttx, ηtxx, ηtxt, ηxtt, ηxtx, ηxxt, ηxxx),
(B.26)
where for smooth section, ηtx = ηxt, ηtxx = ηxtx = ηxxt, · · · . The Lagrangian
density along the third jet of a section φ can be given by
L(j3(φ)) = 12
(
ηxη2t +
η2tx
ηx
+ (ηxηtxx − ηtxηxx)
2
η5x
)
dx ∧ dt. (B.27)
Because the Lagrangian density depends only on ηt, ηx, ηtx, ηxx, ηtxx, so the
Cartan form θL = (FL)∗θ can be found by direct calculation or from Theorem
B.7 (from now on we change the notion η to y):
θL = T1dx ∧ dt+ T2 + T3 + T4 (B.28)
where
T1 = L−
∂L
∂yt
yt −
∂L
∂yx
yx +Dx
( ∂L
∂ytx
)
yt +Dx
( ∂L
∂yxx
)
yx
− ∂L∂ytx
ytx −
∂L
∂yxx
yxx −Dxx
( ∂L
∂ytxx
)
yt +Dx
( ∂L
∂ytxx
)
ytx −
∂L
∂ytxx
ytxx.
T2 = p1dy ∧ dt− p0dy ∧ dx,
T3 = p01dyt ∧ dt + p11dyx ∧ dt,
T4 = p011dytx ∧ dt,
and
p0 = pt = ∂L∂yt
−Dx
( ∂L
∂ytx
)
+Dxx
( ∂L
∂ytxx
)
,
p1 = px = ∂L∂yx
−Dx
( ∂L
∂yxx
)
,
p01 = ptx = ∂L∂ytx
−Dx
( ∂L
∂ytxx
)
,
p11 = pxx = ∂L∂yxx
,
p011 = ptxx = ∂L∂ytxx
,
p10 = p00 = p000 = p110 = p010 = p001 = p100 = p111 = p101 = 0.
130
APPENDIX B. MULTI-SYMPLECTIC FORMULATION B.2. FORMULATION FOR MCH
We can think p0, p1 as the temporal and spatial conjugate momentum of
the field component y, and p01, p11 as the temporal and spatial conjugate
momentum of the field component yx...
The corresponding 3-form ΩL = dθL is
ΩL = dT1 ∧ dx ∧ dt + dp1 ∧ dy ∧ dt− dp0 ∧ dy ∧ dx+ dp01 ∧ dyt ∧ dt+
dp11 ∧ dyx ∧ dt+ dp011 ∧ dytx ∧ dt
(B.29)
and if we introduce a transform
FL : (y, yx, yt, yxx, yxt, ytx, ytt, yxxx, · · · ) 7→ (y, yx, yt, ytx, p1, p0, p01, p11, p011)
from the space of the vertical sections of J5(Y ) 7→ X into the phase space
M = R9 modeled over X = R2, then we have the following
Proposition B.10 For any piXY vertical vectors V,W in F defined in the
formula (B.20), we have
∂
∂xω
1(TFL · j5(V ), TFL · j5(W )) + ∂∂tω
0(TFL · j5(V ), TFL · j5(W )) = 0.
(B.30)
Moreover, the mCH equation (B.22) is equivalent to the Hamiltonian system
of equations on the multi-symplectic structure
B1Zx +B0Zt = ∇H, (B.31)
with the Hamiltonian defined by
H = L− pxyx − ptyt − p01ytx − p11yxx − p011ytxx, (B.32)
where ωi, Bi are defined as follows: ωi(u, v) = vTBiu for any u, v ∈ R9, i =
0, 1 and
B0 =


0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
−1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0


,
131
APPENDIX B. MULTI-SYMPLECTIC FORMULATION B.2. FORMULATION FOR MCH
B1 =


0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 1
−1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 −1 0 0 0 0 0 0
0 −1 0 0 0 0 0 0 0
0 0 0 −1 0 0 0 0 0


.
Proof The idea is the same as that in the [66]. The only difference consists
of the calculations: for any piXY vertical vectors V,W in F defined in the
formula (B.20), we have
j5(W ) a j5(V ) a ΩL =
(
V p1W y −W p1V y + V p01W yt −W p01V yt
)
dt
+
(
V p011W ytx −W p011V ytx + V p11W yx −W p11V yx
)
dt
−
(
V p0W y −W p0V y
)
dx.
(B.33)
Then the fact that the multi-symplectic formula in Theorem (B.9) holds for
arbitrary ∂Ux and Stokes theorem imply
∂
∂xω
1(Zt, Zx) +
∂
∂tω
0(Zt, Zx) = 0 (B.34)
where ωi and Bi are defined in the Proposition.
132
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