Uniform Regularity for Incompressible MHD Equations in a Bounded Domain with Curved Boundary in 3D

TL;DR

This paper establishes uniform regularity results for incompressible MHD equations in 3D bounded domains with curved boundaries, enabling analysis of the vanishing dissipation limit without requiring equal viscosity and magnetic diffusion coefficients.

Contribution

It develops a novel approach to handle anisotropic regularity and boundary curvature effects in MHD equations with general boundary conditions.

Findings

Proved uniform regularity of conormal Sobolev and Lipschitz norms.

Demonstrated vanishing dissipation limit as viscosity and magnetic diffusion tend to zero.

Overcame challenges posed by boundary curvature and unequal diffusion coefficients.

Abstract

For the initial boundary problem of the incompressible MHD equations in a bounded domain with general curved boundary in 3D with the general Navier-slip boundary conditions for the velocity field and the perfect conducting condition for the magnetic field, we establish the uniform regularity of conormal Sobolev norms and Lipschitz norms to addressing the anisotropic regularity of tangential and normal directions, which enable us to prove the vanishing dissipation limit as the viscosity and the magnetic diffusion coefficients tend to zero. We overcome the difficulties caused by the intricate interaction of boundary curvature, velocity field, and magnetic fields and resolve the issue caused by the problem that the viscosity and the magnetic diffusion coefficients are not required to be equal.