Zero kinematic viscosity-magnetic diffusion limit of the incompressible viscous magnetohydrodynamic equations with Navier boundary conditions

TL;DR

This paper studies the behavior of solutions to the incompressible viscous magnetohydrodynamic equations as both the kinematic viscosity and magnetic diffusion coefficients approach zero, establishing uniform bounds and convergence rates.

Contribution

It provides the first uniform regularity results and convergence rates for the zero viscosity-magnetic diffusion limit under Navier boundary conditions.

Findings

Solutions are uniformly bounded in conormal Sobolev and $W^{1,inity}$ spaces.

Established convergence rates as viscosity and magnetic diffusion tend to zero.

Achieved uniform regularity independent of the small parameters.

Abstract

We investigate the zero kinematic viscosity-magnetic diffusion limit of the incompressible viscous magnetohydrodynamic equations with Navier boundary conditions in a smooth bounded domain $\Omega\subset\mathbb{R}^3$. We obtain the uniform regularity of solutions with respect to the kinematic viscosity coefficient and the magnetic diffusivity coefficient. These solutions are uniformly bounded in a conormal Sobolev space and $W^{1,\infty}(\Omega)$ which allow us to take the zero kinematic viscosity-magnetic diffusion limit. Moreover, we also get the rates of convergence.