Semi-classical limit of the massive Klein-Gordon-Maxwell system toward the relativistic Euler-Maxwell system via an adapted modulated energy method

TL;DR

This paper demonstrates the convergence of solutions from the massive Klein-Gordon-Maxwell system to the relativistic Euler-Maxwell system in the semi-classical limit, using a modulated energy approach.

Contribution

It introduces an adapted modulated energy method to analyze the semi-classical limit and establishes well-posedness and connections to Vlasov-Maxwell equations.

Findings

Convergence of momentum, density, and electromagnetic field in the semi-classical limit.

Proof of well-posedness for the relativistic Euler-Maxwell system.

Relation between Klein-Gordon-Maxwell, Euler-Maxwell, and Vlasov-Maxwell systems.

Abstract

We show that the momentum, the density, and the electromagnetic field associated with the massive KleinGordon-Maxwell equations converge in the semi-classical limit towards their respective equivalents associated with the relativistic Euler-Maxwell equations. The proof relies on a modulated stress-energy method and a compactness argument. We also give a proof of the well-posedness of the relativistic Euler-Maxwell equations and show how this system, and so the semi-classical limit of Klein-Gordon-Maxwell, is related to the relativistic massive Vlasov-Maxwell equations.