| | |
| | |
Stat |
Members: 3645 Articles: 2'506'133 Articles rated: 2609
27 April 2024 |
|
| | | |
|
Article overview
| |
|
Improved decay rates with small regularity loss for the wave equation about a Schwarzschild black hole | P. Blue
; A. Soffer
; | Date: |
7 Dec 2006 | Subject: | Analysis of PDEs | Abstract: | We continue our study of the decoupled wave equation in the exterior of a spherically symmetric, Schwarzschild, black hole. Because null geodesics on the photon sphere orbit the black hole, extra effort must be made to show that the high angular momentum components of a solution decay sufficiently fast, particularly for low regularity initial data. Previous results are rapid decay for regular ($H^3$) initial data cite{BSterbenz} and slower decay for rough ($H^{1+epsilon}$) initial data cite{BlueSoffer3}. Here, we combine those methods to show boundedness of the conformal charge. From this, we conclude that there are bounds for global in time, space-time norms, in particular int_I ildephi ^4 d^4vol < C for $H^{1+epsilon}$ initial data with additional decay towards infinite and the bifurcation sphere. Here $ ildephi$ refers to a solution of the wave equation. $I$ denotes the exterior region of the Schwarzschild solution, which can be expressed in coordinates as $r>2M$, $tinReals, omegain S^2$, and $d^4 ext{vol}$ is the natural 4-dimensional volume induced by the Schwarzschild pseudo-metric. We also demonstrate that the photon sphere has the same influence on the wave equation as a closed geodesic has on the wave equation on a Riemannian manifold. We demonstrate this similarity by extending our techniques to the wave equation on a class of Riemannian manifolds. Under further assumptions, the space-time estimates are sufficient to prove global bounds for small data, nonlinear wave equations on a class of Riemannian manifolds with closed geodesics. We must use global, space-time integral estimates since $L^infty$ estimates cannot hold at this level of regularity. | Source: | arXiv, math/0612168 | Services: | Forum | Review | PDF | Favorites |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)
|
| |
|
|
|
| News, job offers and information for researchers and scientists:
| |