Axisymmetric Incompressible Viscous Plasmas: Global Well-Posedness and Asymptotics

TL;DR

This paper proves the global well-posedness of 3D axisymmetric viscous plasma equations with Maxwell's laws for large initial data and analyzes the asymptotic limit as the speed of light tends to infinity, deriving the viscous MHD system.

Contribution

It establishes the first global well-posedness result for 3D viscous plasmas with full Maxwell equations and analyzes the singular limit to MHD in the axisymmetric setting.

Findings

Global solutions exist for large initial data when c is sufficiently large.

Solutions converge to the viscous MHD system as c approaches infinity.

Uniform energy estimates are obtained that are independent of c.

Abstract

This paper is devoted to the global analysis of the three-dimensional axisymmetric Navier--Stokes--Maxwell equations. More precisely, we are able to prove that, for large values of the speed of light $c\in (c_0, \infty)$, for some threshold $c_0>0$ depending only on the initial data, the system in question admits a unique global solution. The ensuing bounds on the solutions are uniform with respect to the speed of light, which allows us to study the singular regime $c\rightarrow \infty$ and rigorously derive the limiting viscous magnetohydrodynamic (MHD) system in the axisymmetric setting. The strategy of our proofs draws insight from recent results on the two-dimensional incompressible Euler--Maxwell system to exploit the dissipative--dispersive structure of Maxwell's system in the axisymmetric setting. Furthermore, a detailed analysis of the asymptotic regime $c\to\infty$ allows us to derive a robust nonlinear energy estimate which holds uniformly in $c$. As a byproduct of such refined uniform estimates, we are able to describe the global strong convergence of solutions toward the MHD system. This collection of results seemingly establishes the first available global well-posedness of three-dimensional viscous plasmas, where the electric and magnetic fields are governed by the complete Maxwell equations, for large initial data as $c\to\infty$.