Dynamics and numerics of generalised Euler equations : a thesis submitted to Massey University in partial fulfillment of the requirements for the degree of Ph.D. in Mathematics, Palmerston North, New Zealand
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 H[superscript]k metric , k[is greater than or equal to]2.
The term “generalised Euler equations” is used to describe geodesic equations
on Lie groups, which unifies many differential equations and has found
many applications in such as hydrodynamics, medical imaging in the computational
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[is greater than or equal to]2, which we call the modified
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 [blackboard bold] 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
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 numerical
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
and obtain the short time existence and well-posedness for the generalised
Euler equation in the two dimension case.