From resolvent estimates to damped waves

2012/06/07 by Hans Christianson, Christianson, Hans, Emmanuel Schenck +5 · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory (math.SP) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1206.1565

openalex publication_date 2012/06/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

Abstract

In this paper we show how to obtain decay estimates for the damped wave equation on a compact manifold without geometric control via knowledge of the dynamics near the un-damped set. We show that if replacing the damping term with a higher-order complex absorbing potential gives an operator enjoying polynomial resolvent bounds on the real axis, then the "resolvent" associated to our damped problem enjoys bounds of the same order. It is known that the necessary estimates with complex absorbing potential can also be obtained via gluing from estimates for corresponding non-compact models.

Citations

• Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter\n spaces

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• Geometry of wave damping on the torus

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