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Dispersive and Strichartz estimates for 3D wave equation with a class of many-electric potentials

Haoran Wang

Abstract

We prove the dispersive and Strichartz estimates for solutions to the wave equation with a class of many-electric potentials in spatial dimension three. To obtain the desired dispersive estimate, based on the spectral properties of the Schrödinger operator involved, we subsequently prove the dispersive estimate for the corresponding Schrödinger semigroup, obtain a Gaussian type upper bound, establish Bernstein type inequalities, and finally pass to the Müller-Seeger's subordination formula. The desired Strichartz estimates follow by the established dispersive estimate and the standard argument of Keel-Tao.

Dispersive and Strichartz estimates for 3D wave equation with a class of many-electric potentials

Abstract

We prove the dispersive and Strichartz estimates for solutions to the wave equation with a class of many-electric potentials in spatial dimension three. To obtain the desired dispersive estimate, based on the spectral properties of the Schrödinger operator involved, we subsequently prove the dispersive estimate for the corresponding Schrödinger semigroup, obtain a Gaussian type upper bound, establish Bernstein type inequalities, and finally pass to the Müller-Seeger's subordination formula. The desired Strichartz estimates follow by the established dispersive estimate and the standard argument of Keel-Tao.
Paper Structure (10 sections, 12 theorems, 151 equations)

This paper contains 10 sections, 12 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. preliminaries
  3. dispersive estimate for Schrödinger flow
  4. heat kernel
  5. Bernstein inequalities, Besov spaces & subordination formula
  6. Bernstein inequalities
  7. Besov spaces
  8. Subordination formula
  9. proof of Theorem \ref{['thm:wave']}
  10. proof of Theorem \ref{['thm:str']}

Key Result

Theorem 1.1

Let $T$ be a constant in $(0,\pi)$, then, for any $t\in(0,T)$, there exists a positive constant $C$ independent of $t$ such that, for all $f\in\dot{\mathcal{B}}^{3/2}_{1,1}(\mathbb{R}^3)$ and $g\in\dot{\mathcal{B}}^{1}_{1,1}(\mathbb{R}^3)$

Theorems & Definitions (32)

  • Theorem 1.1: Dispersive estimate
  • Remark 1.2
  • Theorem 1.3: Strichartz estimate
  • Remark 1.4
  • Lemma 2.1
  • Remark 2.2
  • proof : Proof of Lemma \ref{['lem:sum-ker']}
  • Proposition 3.1: Schrödinger kernel
  • Remark 3.2
  • proof : Proof of Proposition \ref{['prop:S-ker']}
  • ...and 22 more