Mathematical aspects of the cold plasma model

TL;DR

This paper analyzes a simplified electromagnetic wave propagation model in zero-temperature plasma, revealing mathematical challenges and conditions for well-posed boundary value problems, and shedding light on general issues in mixed elliptic-hyperbolic PDEs.

Contribution

It provides a mathematical analysis of the cold plasma model, highlighting ill-posedness issues and conditions for solution existence in boundary value problems.

Findings

Boundary value problems are ill-posed under conventional electromagnetic theory.

Solutions can exist in a weak sense under certain prescribed conditions.

The analysis sheds light on difficulties in solving mixed elliptic-hyperbolic PDEs.

Abstract

A simple model for electromagnetic wave propagation through zero-temperature plasma is analyzed. Many of the complexities of the plasma state are present even under these idealized conditions, and a number of mathematical difficulties emerge. In particular, boundary value problems formulated on the basis of conventional electromagnetic theory turn out to be ill-posed in this context. However, conditions may be prescribed under which solutions to the Dirichlet problem exist in an appropriately weak sense. In addition to its physical interest, analysis of the cold plasma model illuminates generic difficulties in formulating and solving boundary value problems for mixed elliptic-hyperbolic partial differential equations.