On the role of the displacement current and the Cattaneo's law on boundary layers of plasma

TL;DR

This paper analyzes how the displacement current and Cattaneo's law influence boundary layers in relativistic plasma flows, revealing new hyperbolic boundary layer structures and conditions for their physical and mathematical stability.

Contribution

It introduces a new family of hyperbolic boundary layers in plasma models and establishes conditions for their well-posedness and stability, considering the effects of displacement current and Cattaneo's law.

Findings

New hyperbolic boundary layer solutions identified.

Conditions under which boundary layers resemble Prandtl or Hartmann layers.

Cattaneo's law and displacement current may stabilize the system.

Abstract

In the present paper, we aim to mathematically analyse the role of the displacement current and the Cattaneo's law on the boundary-layer theory of plasma, when the corresponding characteristic speed is relativistic. We restrict our analysis to two-dimensional flows and we study the asymptotic limit of the Navier-Stokes-Maxwell equations with Cattaneo's law near a bounding flat line, when the Hartmann, Reynolds and magnetic Reynolds numbers proportionally diverge to infinity. The goal of this paper is twofold. We first show that the extended version of the Navier-Stokes-Maxwell equations leads to a new family of boundary layers, which are hyperbolic both on the momentum equation and the Ampere's law. Secondly, we address the well-posedness of the derived equations and show the existence of global-in-time analytic solutions for small initial data. Our modelling highlights which conditions on the dimensionless parameters allow to interpret the proposed system as boundary layers with thickness typical of Prandtl or Hartmann. Furthermore, our development shows that the conditions related to Hartmann might be more physically acceptable. Finally, our analysis suggests that the Cattaneo's law and the displacement current might indeed stabilise the derived system in terms of existence of global-in-time analytic solutions.