Semi-classical limit for Klein-Gordon equation toward relativistic Euler equations via an adapted modulated energy method

TL;DR

This paper proves the convergence of solutions from the massive nonlinear Klein-Gordon equation to the relativistic Euler system in the semi-classical limit, using an adapted modulated energy method for wave equations.

Contribution

It introduces a novel modulated stress-energy method tailored for the wave and relativistic equations to establish this convergence.

Findings

Klein-Gordon solutions converge to relativistic Euler solutions in Lebesgue norms.

Momentum and density of Klein-Gordon match those of the Euler system in the limit.

The method applies to the relativistic setting with potential, extending previous techniques.

Abstract

We show the convergence of the solutions to the massive nonlinear Klein-Gordon equation toward solutions to a relativistic Euler with potential type system in the semi-classical limit. In particular, the momentum and the density of Klein-Gordon converge to the the momentum and the density of the relativistic Euler system in Lebesgue norms. The relativistic Euler with potential is equivalent to the usual relativistic Euler with pressure up to a rescaling. The proof relies on the modulated energy method adapted to the wave equation and the relativistic setting: a modulated stress-energy method.