On the Yudovich solutions for the ideal MHD equations

Taoufik Hmidi

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper, we address the problem of weak solutions of Yudovich type for the inviscid magnetohydrodynamic (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.

Original languageEnglish (US)
Pages (from-to)3117-3158
Number of pages42
JournalNonlinearity
Volume27
Issue number12
DOIs
StatePublished - Dec 1 2014

Keywords

  • Inviscid MHD equations
  • Potential theory
  • Vortex patches

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

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