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Dispersive estimates for a system of tensorial quasilinear wave equations satisfying the weak-null condition

Sari Ghanem

Abstract

We establish both global existence and decay properties for solutions with small data for a general class of coupled system of tensorial quasilinear hyperbolic wave equations in three space dimensions, that covers the dynamical Einstein equations coupled to a class of non-linear matter sources that do not satisfy the null condition of Christodoulou and Klainerman, and have new different non-linearities than the one treated by Lindblad-Rodnianski, for which their celebrated seminal $L^\infty$-estimate does not work, to the best of our knowledge. Global existence of solutions for a general class of quasilinear wave equations satisfying the weak-null condition, with small initial data, is largely an open problem at present. There is no known theory to prove decay for the class of non-linear hyperbolic partial differential equations that we treat in this paper. We establish a technique based on novel decoupling of the higher order energy estimates, at the level of the $L^2$-norm of the Lie derivatives of the tangential components, without involving all the other components, up to some good factor. This generalizes our previous results to include new non-linearities that are not present in the Einstein-Yang-Mills system in the Lorenz gauge.

Dispersive estimates for a system of tensorial quasilinear wave equations satisfying the weak-null condition

Abstract

We establish both global existence and decay properties for solutions with small data for a general class of coupled system of tensorial quasilinear hyperbolic wave equations in three space dimensions, that covers the dynamical Einstein equations coupled to a class of non-linear matter sources that do not satisfy the null condition of Christodoulou and Klainerman, and have new different non-linearities than the one treated by Lindblad-Rodnianski, for which their celebrated seminal -estimate does not work, to the best of our knowledge. Global existence of solutions for a general class of quasilinear wave equations satisfying the weak-null condition, with small initial data, is largely an open problem at present. There is no known theory to prove decay for the class of non-linear hyperbolic partial differential equations that we treat in this paper. We establish a technique based on novel decoupling of the higher order energy estimates, at the level of the -norm of the Lie derivatives of the tangential components, without involving all the other components, up to some good factor. This generalizes our previous results to include new non-linearities that are not present in the Einstein-Yang-Mills system in the Lorenz gauge.
Paper Structure (27 sections, 20 theorems, 159 equations)

This paper contains 27 sections, 20 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Novelty of the current work and comparison with the previous studies
  3. Novelty
  4. Related works
  5. The coupled dynamical system of tensorial quasilinear wave equations
  6. The initial data set and the norms
  7. The condition on the fixed system of coordinates
  8. The statement of the theorem
  9. Strategy of the proof
  10. The continuity argument
  11. The recursive formula of Grönwall type inequalities
  12. The idea of the proof
  13. The decoupled higher order energy estimates
  14. À priori pointwise decay on the fields
  15. The decoupled energy estimates for the tangential components
  16. ...and 12 more sections

Key Result

Theorem 1.1

We consider in a certain system of coordinates that satisfies the condition wavecoordinatesestimateonLiederivativesZonmetric, which is the case for wave coordinates, the dynamical tensorial system of coupled quasilinear wave equations badtensorialcoupledwaveequation-goodtensorialcoupledwaveequation, then, we have for all time $t$ , in the entire exterior region, Furthermore, for all $|I| \leq K -

Theorems & Definitions (68)

  • Definition 1.0.1
  • Definition 1.0.2
  • Remark 1.0.3
  • Definition 1.0.4
  • Definition 1.0.5
  • Definition 1.0.6
  • Remark 1.0.7
  • Definition 1.2.1
  • Remark 1.2.2
  • Definition 1.2.3
  • ...and 58 more