Existence and Uniqueness of Singular Solutions for a Conservation Law Arising in Magnetohydrodynamics

TL;DR

This paper addresses the existence and uniqueness of singular solutions for a magnetohydrodynamics conservation law system by introducing a nonlinear change of variables and an admissibility criterion, enabling standard solution methods.

Contribution

It introduces a nonlinear transformation and an admissibility criterion that ensure existence and uniqueness of singular solutions for the MHD system.

Findings

A nonlinear change of variables simplifies the Riemann problem.

An admissibility criterion guarantees uniqueness of solutions.

Singular solutions with Dirac masses are well-defined and unique.

Abstract

The Brio system is a two-by-two system of conservation laws arising as a simplified model in ideal magnetohydrodynamics (MHD). The system has the form \begin{align*} \partial_t u+\partial_x \Big({\textstyle \frac{u^2+v^2}{2}}\Big)=0,\\ \partial_t v+\partial_x \big(v(u-1)\big)=0. \end{align*} It was found in previous works that the standard theory of hyperbolic conservation laws does not apply to this system since the characteristic fields are not genuinely nonlinear on the set $v=0$. As a consequence, certain Riemann problems have no weak solutions in the traditional class of functions of bounded variation. It was argued in Nonlinearity 9, 1547--1563 (1996) that in order to solve the system, singular solutions containing Dirac masses along the shock waves might have to be used. Solutions of this type were exhibited in Proc. Edinb. Math. Soc. 55, 711--729 (2012) and Russ. J. Math. Phys. 22, 518--527 (2015), but uniqueness was not obtained. In the current work, we introduce a nonlinear change of variables which makes it possible to solve the Riemann problem in the framework of the standard theory of conservation laws. In addition, we develop a criterion which leads to an admissibility condition for singular solutions of the original system, and it can be shown that admissible solutions are unique in the framework developed here.