Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Tony Salvi

TL;DR

The work analyzes the semi-classical limit of the massive nonlinear Klein–Gordon equation with a polynomial defocusing potential on Minkowski space, proving convergence of KG momentum and density to a relativistic Euler with potential (REP) system using a modulated stress–energy approach. By constructing a modulated energy that measures the discrepancy between KG and REP dynamics, the authors establish coercivity and propagation properties, obtaining quantitative convergence rates in Lebesgue spaces and, for the critical case $\gamma=2$, strong time-uniform convergence. They connect REP to the standard relativistic Euler with pressure, providing explicit relations between densities, velocities, and energy densities, and discuss non-relativistic limits. The results illuminate how high-frequency relativistic waves funnel into relativistic hydrodynamics, with potential extensions to other spacetime geometries and single-direction oscillations.

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.

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

TL;DR

The work analyzes the semi-classical limit of the massive nonlinear Klein–Gordon equation with a polynomial defocusing potential on Minkowski space, proving convergence of KG momentum and density to a relativistic Euler with potential (REP) system using a modulated stress–energy approach. By constructing a modulated energy that measures the discrepancy between KG and REP dynamics, the authors establish coercivity and propagation properties, obtaining quantitative convergence rates in Lebesgue spaces and, for the critical case , strong time-uniform convergence. They connect REP to the standard relativistic Euler with pressure, providing explicit relations between densities, velocities, and energy densities, and discuss non-relativistic limits. The results illuminate how high-frequency relativistic waves funnel into relativistic hydrodynamics, with potential extensions to other spacetime geometries and single-direction oscillations.

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.
Paper Structure (21 sections, 22 theorems, 125 equations)

This paper contains 21 sections, 22 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Context on the Klein Gordon equation
  3. WKB approximation
  4. First version of the theorem
  5. Idea of the proof
  6. Acknowledgments
  7. Outline of the paper
  8. Notations
  9. Klein-Gordon equation
  10. Relativistic Euler with potential
  11. Definition of the new modulated energy
  12. Main results
  13. Proof of the Theorem \ref{['unTheorem:TH1mainth']}
  14. Existence and regularity of the solutions
  15. Coercivity property, point \ref{['item:modpoint1Idea']}
  16. ...and 6 more sections

Key Result

Proposition 1.1

Let $\Phi_1^\varepsilon=e^{i\frac{\omega}{\varepsilon}}A$ with $\omega$ a smooth real phase and $A$ a smooth complex amplitude satisfying Then, $\Phi_1^\varepsilon$ is an approximate solution of order $2$ to eq:KGPintro, that is

Theorems & Definitions (54)

  • Proposition 1.1
  • proof
  • Remark 1.2
  • Proposition 1.3
  • proof
  • Remark 1.4
  • Theorem 1.1
  • Theorem 1.2
  • Definition 3.1
  • Definition 3.2
  • ...and 44 more