2014/01/24 by Hmidi Taoufik, Taoufik, Hmidi · 1 citation
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Boundary (topology) #Boundary value problem #Classical mechanics #Compressibility #Constraint (computer-aided design) #Euler equations #Euler's formula #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry #Ideal (ethics) #Inviscid flow #Magnetic field #Magnetohydrodynamics #Mathematical analysis #Mathematics #Mechanics #Navier-Stokes equation solutions #Physics #Uniqueness #Vortex #Vorticity #math.AP
paper · pdf · doi:10.48550/arxiv.1401.6326
published in arXiv (Cornell University) (Cornell University) · 40 pages
arxiv created 2014/01/24 · openalex publication_date 2014/01/24 · arxiv updated 2014/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.