Dispersive estimates for a system of tensorial quasilinear wave equations satisfying the weak-null condition
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| Format: | Preprint |
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2026
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| author | Ghanem, Sari |
| author_facet | Ghanem, Sari |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_01939 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dispersive estimates for a system of tensorial quasilinear wave equations satisfying the weak-null condition Ghanem, Sari Analysis of PDEs General Relativity and Quantum Cosmology Differential Geometry 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. |
| title | Dispersive estimates for a system of tensorial quasilinear wave equations satisfying the weak-null condition |
| topic | Analysis of PDEs General Relativity and Quantum Cosmology Differential Geometry |
| url | https://arxiv.org/abs/2603.01939 |