Cubic quasilinear wave equation and bilinear estimates

Bahouri, Hajer; Chemin, Jean-Yves

doi:10.1090/conm/350/06335

Cited by 2 publications

(2 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Low-regularity problems are interesting for quasilinear wave equations beyond the Einstein equations, see for example [26,64,70]. We highlight particularly the work [5] of Bahouri-Chemin on the high-dimensional low-regularity well-posedness of a coupled wave-elliptic system similar to the structure of the polarized U(1)-reduced Einstein vacuum equations in an elliptic gauge.…”

Section: Low-regularity Problems In General Relativity and Beyondmentioning

confidence: 97%

See 1 more Smart Citation

Nonlinear interaction of three impulsive gravitational waves I: main result and the geometric estimates

Luk,

Van de Moortel

2021

Preprint

View full text Add to dashboard Cite

Impulsive gravitational waves are (weak) solutions to the Einstein vacuum equations such that the Riemann curvature tensor admits a delta singularity along a null hypersurface. The interaction of impulsive gravitational waves is then represented by the transversal intersection of these singular null hypersurfaces. This is the first of a series of two papers in which we prove that for all suitable U(1)-symmetric initial data representing three "small amplitude" impulsive gravitational waves propagating towards each other transversally, there exists a local solution to the Einstein vacuum equations featuring the interaction of these waves. Moreover, we show that the solution remains Lipschitz everywhere and is H 2 loc ∩ C1, 1 4 − loc away from the impulsive gravitational waves. This is the first construction of solutions to the Einstein vacuum equations featuring the interaction of three impulsive gravitational waves.In this paper, we focus on the geometric estimates, i.e. we control the metric and the null hypersurfaces assuming the wave estimates. The geometric estimates rely crucially on the features of the spacetime with three interacting impulsive gravitational waves, particularly that each wave is highly localized and that the waves are transversal to each other. In the second paper of the series, we will prove the wave estimates and complete the proof.

show abstract

Section: Low-regularity Problems In General Relativity and Beyondmentioning

confidence: 97%

“…For U(1) symmetry (even without polarization), the results of [64] imply that local well-posedness can be obtained for in H 7 4 + . While the optimal regularity in polarized U(1) symmetry is not explicitly discussed in the literature, some interesting progress has been made on a related quasilinear model problem [5,33].…”

Section: Introductionmentioning

confidence: 99%