On the vanishing dissipation limit for the incompressible MHD equations on bounded domains

TL;DR

This paper studies the behavior of solutions to 3D viscous MHD equations in bounded domains, focusing on existence, regularity, and the limit as dissipation vanishes, with results on convergence to ideal MHD solutions.

Contribution

It establishes global weak solutions in general domains and uniform local well-posedness with convergence rates in flat domains as dissipation approaches zero.

Findings

Existence of global weak solutions in smooth bounded domains.

Uniform local well-posedness with higher regularity in flat domains.

Convergence of viscous solutions to ideal MHD solutions as dissipation vanishes.

Abstract

In this paper, we investigate the solvability, regularity and the vanishing dissipation limit of solutions to the three-dimensional viscous magneto-hydrodynamic (MHD) equations in bounded domains. On the boundary, the velocity field fulfills a Navier-slip condition, while the magnetic field satisfies the insulating condition. It is shown that the initial-boundary problem has a global weak solution for a general smooth domain. More importantly, for a flat domain, we establish the uniform local well-posedness of the strong solution with higher order uniform regularity and the asymptotic convergence with a rate to the solution of the ideal MHD as the dissipation tends to zero.