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Global solutions of quasilinear systems of Klein--Gordon equations in 3D

Alexandru D. Ionescu, Benoit Pausader

TL;DR

The paper addresses global well-posedness and scattering for small, localized data in three-dimensional quasilinear Klein–Gordon systems with different propagation speeds, establishing explicit nondegeneracy conditions that guarantee global solutions. The authors develop a robust Fourier-analytic framework combining energy estimates with a two-component Z-norm to manage both decay and large space–time resonances, enabling time-integrable bounds in L∞ despite resonant interactions. They apply the method to the Euler–Maxwell system for electrons, proving robust global stability of the neutral equilibrium under small perturbations, with explicit dependence on problem parameters. The approach advances the understanding of multispeed dispersive systems by providing a parameter-explicit, resonance-aware bootstrap that remains effective in the presence of large resonance sets in 3D.

Abstract

We prove small data global existence and scattering for quasilinear systems of Klein-Gordon equations with different speeds, in dimension three. As an application, we obtain a robust global stability result for the Euler-Maxwell equations for electrons.

Global solutions of quasilinear systems of Klein--Gordon equations in 3D

TL;DR

The paper addresses global well-posedness and scattering for small, localized data in three-dimensional quasilinear Klein–Gordon systems with different propagation speeds, establishing explicit nondegeneracy conditions that guarantee global solutions. The authors develop a robust Fourier-analytic framework combining energy estimates with a two-component Z-norm to manage both decay and large space–time resonances, enabling time-integrable bounds in L∞ despite resonant interactions. They apply the method to the Euler–Maxwell system for electrons, proving robust global stability of the neutral equilibrium under small perturbations, with explicit dependence on problem parameters. The approach advances the understanding of multispeed dispersive systems by providing a parameter-explicit, resonance-aware bootstrap that remains effective in the presence of large resonance sets in 3D.

Abstract

We prove small data global existence and scattering for quasilinear systems of Klein-Gordon equations with different speeds, in dimension three. As an application, we obtain a robust global stability result for the Euler-Maxwell equations for electrons.

Paper Structure

This paper contains 22 sections, 26 theorems, 526 equations.

Table of Contents

  1. Introduction
  2. Statement of the results
  3. Comments and plan of the proof
  4. Previous results on systems of Klein-Gordon equations.
  5. General strategy.
  6. Space-time resonant sets.
  7. Norms.
  8. Condition on the parameters.
  9. Reductions and proofs of the main theorems
  10. Local existence results
  11. Definitions, function spaces, and the main propositions
  12. Proof of Theorem \ref{['MainThm1']}
  13. Proof of Theorem \ref{['MainThm2']}
  14. Proof of Proposition \ref{['Norm0']}
  15. Proof of Proposition \ref{['Norm']}
  16. ...and 7 more sections

Key Result

Theorem 1.1

Assume $A\geq 1$, $d\geq 1$, and $b_1,\ldots,b_d,c_1,\ldots,c_d\in[1/A,A]$ satisfy the non-resonance conditions We fix quadratic nonlinearities $(F_{\mu})_{\mu\in\{1,\ldots,d\}}$ as in KGNon1--CondForEE, let $N_0=10^4$, and assume that $v_0,v_1:\mathbb{R}^3\to\mathbb{R}^d$ satisfy the smallness conditions where $\overline{\varepsilon}=\overline{\varepsilon}(d,A,F_\mu)>0$ is sufficiently small (d

Theorems & Definitions (48)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof : Proof of Proposition \ref{['Localexistence0']}
  • Proposition 2.2
  • proof : Proof of Proposition \ref{['Localexistence']}
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • ...and 38 more