Massive wave propagation near null infinity
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913623944724480 |
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| author | Sussman, Ethan |
| author_facet | Sussman, Ethan |
| contents | We study, fully microlocally, the propagation of massive waves on the octagonal compactification \[\mathbb{O}=[\overline{\mathbb{R}^{1,d}};\mathscr{I};1/2]\] of asymptotically Minkowski spacetime, which allows a detailed analysis both at timelike and spacelike infinity (as previously investigated using Parenti-Shubin-Melrose's sc-calculus) and, more novelly, at null infinity, denoted $\mathscr{I}$. The analysis is closely related to Hintz-Vasy's recent analysis of massless wave propagation at null infinity using the ``e,b-calculus'' on $\mathbb{O}$. We prove several elementary corollaries regarding the Klein-Gordon IVP. Our main technical tool is a fully symbolic pseudodifferential calculus, $Ψ_{\mathrm{de,sc}}(\mathbb{O})$, the ``de,sc-calculus'' on $\mathbb{O}$. The `de' refers to the structure (``double edge'') of the calculus at null infinity, and the `sc' refers to the structure (``scattering'') at the other boundary faces. We relate this structure to the hyperbolic coordinates used in other studies of the Klein-Gordon equation. Unlike hyperbolic coordinates, the de,sc-boundary fibration structure is Poincaré invariant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_01119 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Massive wave propagation near null infinity Sussman, Ethan Analysis of PDEs Mathematical Physics Primary: 35L05, 35B40. Secondary: 35P25, 35C20, 58J40, 58J47 We study, fully microlocally, the propagation of massive waves on the octagonal compactification \[\mathbb{O}=[\overline{\mathbb{R}^{1,d}};\mathscr{I};1/2]\] of asymptotically Minkowski spacetime, which allows a detailed analysis both at timelike and spacelike infinity (as previously investigated using Parenti-Shubin-Melrose's sc-calculus) and, more novelly, at null infinity, denoted $\mathscr{I}$. The analysis is closely related to Hintz-Vasy's recent analysis of massless wave propagation at null infinity using the ``e,b-calculus'' on $\mathbb{O}$. We prove several elementary corollaries regarding the Klein-Gordon IVP. Our main technical tool is a fully symbolic pseudodifferential calculus, $Ψ_{\mathrm{de,sc}}(\mathbb{O})$, the ``de,sc-calculus'' on $\mathbb{O}$. The `de' refers to the structure (``double edge'') of the calculus at null infinity, and the `sc' refers to the structure (``scattering'') at the other boundary faces. We relate this structure to the hyperbolic coordinates used in other studies of the Klein-Gordon equation. Unlike hyperbolic coordinates, the de,sc-boundary fibration structure is Poincaré invariant. |
| title | Massive wave propagation near null infinity |
| topic | Analysis of PDEs Mathematical Physics Primary: 35L05, 35B40. Secondary: 35P25, 35C20, 58J40, 58J47 |
| url | https://arxiv.org/abs/2305.01119 |