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arXiv · 2003.03114

A semi-discrete scheme derived from variational principles for global conservative solutions of a Camassa-Holm system

Sondre Tesdal Galtung·Xavier Raynaud
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Abstract

We define a kinetic and a potential energy such that the principle of stationary action from Lagrangian mechanics yields a Camassa--Holm system (2CH) as the governing equations. After discretizing these energies, we use the same variational principle to derive a semi-discrete system of equations as an approximation of the 2CH system. The discretizaton is only available in Lagrangian coordinates and requires the inversion of a discrete Sturm--Liouville operator with time-varying coefficients. We show the existence of fundamental solutions for this operator at initial time with appropriate decay. By propagating the fundamental solutions in time, we define an equivalent semi-discrete system for which we prove that there exists unique global solutions. Finally, we show how the solutions of the semi-discrete system can be used to construct a sequence of functions converging to the conservative solution of the 2CH system.

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Sondre Tesdal Galtung, Xavier Raynaud. 2020-03-06. A semi-discrete scheme derived from variational principles for global conservative solutions of a Camassa-Holm system. https://doi.org/10.1088/1361-6544/abc101

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