A note on multi-dimensional Camassa-Holm type systems on the torus

TL;DR

This paper introduces a multi-dimensional nonlinear PDE system encompassing Camassa-Holm and Hunter-Saxton equations, applying geometric methods to analyze well-posedness, conservation laws, and stability, with a focus on the torus setting.

Contribution

It extends geometric analysis to a broad class of multi-dimensional PDEs related to fluid dynamics, including new insights on their geometric structure and properties.

Findings

Establishment of well-posedness results for the system.

Identification of conservation laws within the PDE framework.

Analysis of stability properties of solutions.

Abstract

We present a $2n$-component nonlinear evolutionary PDE which includes the $n$-dimensional versions of the Camassa-Holm and the Hunter-Saxton systems as well as their partially averaged variations. Our goal is to apply Arnold's [V.I. Arnold, Sur la g\'eom\'etrie diff\'erentielle des groupes de Lie de dimension infinie et ses applications \`a l'hydrodynamique des fluides parfaits. Ann. Inst. Fourier (Grenoble) 16 (1966) 319-361], [D.G. Ebin and J.E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. of Math. 92(2) (1970) 102-163] geometric formalism to this general equation in order to obtain results on well-posedness, conservation laws or stability of its solutions. Following the line of arguments of the paper [M. Kohlmann, The two-dimensional periodic $b$-equation on the diffeomorphism group of the torus. J. Phys. A.: Math. Theor. 44 (2011) 465205 (17 pp.)] we present geometric aspects of a two-dimensional periodic $\mu$-$b$-equation on the diffeomorphism group of the torus in this context.