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arXiv · 1401.6326

On the Yudovich solutions for the ideal MHD equations

Hmidi Taoufik
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Abstract

In this paper, we address the problem of weak solutions of Yudovich type for the inviscid MHD equations in two dimensions. The local-in-time existence and uniqueness of these solutions sound to be hard to achieve due to some terms involving Riesz transforms in the vorticity-current formulation. We shall prove that the vortex patches with smooth boundary offer a suitable class of initial data for which the problem can be solved. However this is only done under a geometric constraint by assuming the boundary of the initial vorticity to be frozen in a magnetic field line. We shall also discuss the stationary patches for the incompressible Euler system $(E)$ and the MHD system. For example, we prove that a stationary simply connected patch with rectifiable boundary for the system $(E)$ is necessarily the characteristic function of a disc.

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Hmidi Taoufik. 2014-01-24. On the Yudovich solutions for the ideal MHD equations. https://arxiv.org/abs/1401.6326

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