On short wave-long wave interactions\\ in the relativistic context: Application to the Relativistic Euler Equations

TL;DR

This paper develops a new relativistic model coupling short wave Dirac equations with long wave Euler equations, introduces a novel relativistic Lagrangian transformation, and proves short-time existence and uniqueness of solutions.

Contribution

It introduces a new relativistic short wave-long wave interaction model with a novel Lagrangian transformation and establishes well-posedness results for the system.

Findings

Formulation of a relativistic Lagrangian transformation

Establishment of short-time existence and uniqueness of solutions

Introduction of a coupled Dirac-Euler relativistic model

Abstract

In this paper we introduce a model of relativistic short wave-long wave interaction where the short waves are described by the massless $1+3$-dimensional Thirring model of nonlinear Dirac equation and the long waves are described by the $1+3$-dimensional relativistic Euler equations. The interaction coupling terms are modeled by a potential proportional to the relativistic specific volume in the Dirac equation and an external force proportional to the square modulus of the Dirac wave function in the relativistic Euler equation. An important feature of the model is that the Dirac equations are based on the Lagrangian coordinates of the relativistic fluid flow. In particular, an important contribution of this paper is a clear formulation of the relativistic Lagrangian transformation. This is done by means of the introduction of natural auxiliary dependent variables, rendering the discussion totally similar to the non-relativistic case. As far as the authors know the definition of the Lagrangian transformation given in this paper is new. Finally, we establish the short-time existence and uniqueness of a smooth solution of the Cauchy problem for the regularized model. This follows through the symmetrization of the relativistic Euler equation introduced by Makino and Ukai (1995) and requires a slight extension of a well known theorem of T.~Kato (1975) on quasi-linear symmetric hyperbolic systems.