Geometric scattering for nonlinear wave equations on the Schwarzschild metric
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918396105326592 |
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| author | Xuan, Pham Truong |
| author_facet | Xuan, Pham Truong |
| contents | In this paper, we establish a conformal scattering theory for defocusing semilinear wave equations on Schwarzschild spacetime. We combine the energy and pointwise decay results for solutions obtained in \cite{Yang} with a Sobolev embedding on spacelike hypersurfaces to derive two-sided energy estimates between the energy flux of solutions through the Cauchy initial hypersurface $Σ_0 = \{ t = 0 \}$ and that through the null conformal boundaries $\mathfrak{H}^+ \cup \scri^+$ (respectively, $\mathfrak{H}^- \cup \scri^-$). By combining these estimates with the well-posedness of the Cauchy and Goursat problems for nonlinear wave equations, we construct a bounded linear and locally Lipschitz scattering operator that maps past scattering data to future scattering data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_17745 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometric scattering for nonlinear wave equations on the Schwarzschild metric Xuan, Pham Truong Analysis of PDEs General Relativity and Quantum Cosmology Mathematical Physics Differential Geometry Functional Analysis In this paper, we establish a conformal scattering theory for defocusing semilinear wave equations on Schwarzschild spacetime. We combine the energy and pointwise decay results for solutions obtained in \cite{Yang} with a Sobolev embedding on spacelike hypersurfaces to derive two-sided energy estimates between the energy flux of solutions through the Cauchy initial hypersurface $Σ_0 = \{ t = 0 \}$ and that through the null conformal boundaries $\mathfrak{H}^+ \cup \scri^+$ (respectively, $\mathfrak{H}^- \cup \scri^-$). By combining these estimates with the well-posedness of the Cauchy and Goursat problems for nonlinear wave equations, we construct a bounded linear and locally Lipschitz scattering operator that maps past scattering data to future scattering data. |
| title | Geometric scattering for nonlinear wave equations on the Schwarzschild metric |
| topic | Analysis of PDEs General Relativity and Quantum Cosmology Mathematical Physics Differential Geometry Functional Analysis |
| url | https://arxiv.org/abs/2410.17745 |