Localized energy estimates for perturbed wave equations exterior to non-star-shaped obstacles
Public Deposited- Last Modified
- March 20, 2019
- Creator
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Perry, Jacob
- Affiliation: College of Arts and Sciences, Department of Mathematics
- Abstract
- We establish integrated localized energy estimates for wave equations with small, time-dependent pertubations in exterior domains with Dirichlet boundary conditions in the case where the obstacle can be illuminated from its exterior by a smooth, convex bounded body. This is a generalization of the case of a star-shaped obstacle, for which such estimates are known by prior results. These estimates for perturbed wave equations are particularly useful in applications to quasilinear waves. To establish the estimates, we integrate space-time divergence identities that we develop. In doing so, we are forced to deal with a surface integral on the boundary of our obstacle. We utilize the illuminated coordinate system to produce a multiplier that gives a definite sign in the boundary integral. Such multiplier techniques seem to produce an unsigned lower order error term in the volume integrals, and so the resulting estimate is useful only for solutions that are localized to very high frequencies. With straightforward multiplier techniques unavailable to us to establish a single estimate valid for all frequencies, inspired by recent results, we look separately at the case of very low frequencies and the case of frequencies bounded away from zero. The estimate we prove for very low frequencies relies only on the Dirichlet boundary condition and is valid regardless of the geometry, so no use of the illuminated coordinate system is necessary. We then implement Carleman-type estimates involving exponential weights. Because we are forced to address the surface integral on the boundary of the obstacle, we work in illuminated coordinates, but the weights produced in proving the Carleman estimates give rise to a lower order error term that can be made as small as necessary to establish an estimate for solutions that are frequency localized to any region bounded away from frequency zero. We are then able to appropriately estimate the error terms that arise when we commute frequency cutoffs with our small, time-dependent perturbation.
- Date of publication
- May 2018
- Keyword
- DOI
- Resource type
- Advisor
- Christianson, Hans
- Canzani, Yaiza
- Metcalfe, Jason
- Marzuola, Jeremy
- Taylor, Michael Eugene
- Degree
- Doctor of Philosophy
- Degree granting institution
- University of North Carolina at Chapel Hill Graduate School
- Graduation year
- 2018
- Language
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