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Strichartz Estimates for Wave Equations with Charge Transfer Hamiltonians
About this Title
Gong Chen
Publication: Memoirs of the American Mathematical Society
Publication Year:
2021; Volume 273, Number 1339
ISBNs: 978-1-4704-4974-2 (print); 978-1-4704-6807-1 (online)
DOI: https://doi.org/10.1090/memo/1339
Published electronically: November 3, 2021
Keywords: Strichartz estimates; energy estimate,
local energy decay,
charge transfer model
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries
- 3. Estimates along Slanted Lines
- 4. Endpoint Reversed Strichartz Estimates
- 5. Strichartz Estimates and Energy Bound
- 6. Inhomogeneous Estimates
- 7. Scattering
Abstract
We prove Strichartz estimates (both regular and reversed) for a scattering state to the wave equation with a charge transfer Hamiltonian in $\mathbb {R}^{3}$: \[ \partial _{tt}u-\Delta u+\sum _{j=1}^{m}V_{j}\left (x-\vec {v}_{j}t\right )u=0. \] The energy estimate and the local energy decay of a scattering state are also established. In order to study nonlinear multisoltion systems, we will present the inhomogeneous generalizations of Strichartz estimates and local decay estimates. As an application of our results, we show that scattering states indeed scatter to solutions to the free wave equation. These estimates for this linear models are also of crucial importance for problems related to interactions of potentials and solitons, for example, in [Comm. Math. Phys. 364 (2018), no. 1, pp. 45–82].- Shmuel Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 2, 151–218. MR 397194
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