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Uniform stability of the damped wave equation with a confining potential in the Euclidean space

Antoine Prouff

Abstract

We investigate trend to equilibrium for the damped wave equation with a confining potential in the Euclidean space. We provide with necessary and sufficient geometric conditions for the energy to decay exponentially uniformly. The proofs rely on tools from semiclassical analysis together with the construction of quasimodes of the damped wave operator. In addition to the Geometric Control Condition, which is familiar in the context of compact Riemannian manifolds, our work involves a new geometric condition due to the presence of turning points in the underlying classical dynamics which rules the propagation of waves in the high-energy asymptotics.

Uniform stability of the damped wave equation with a confining potential in the Euclidean space

Abstract

We investigate trend to equilibrium for the damped wave equation with a confining potential in the Euclidean space. We provide with necessary and sufficient geometric conditions for the energy to decay exponentially uniformly. The proofs rely on tools from semiclassical analysis together with the construction of quasimodes of the damped wave operator. In addition to the Geometric Control Condition, which is familiar in the context of compact Riemannian manifolds, our work involves a new geometric condition due to the presence of turning points in the underlying classical dynamics which rules the propagation of waves in the high-energy asymptotics.

Paper Structure

This paper contains 21 sections, 19 theorems, 148 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. Main results
  4. Geometric and dynamical stabilization conditions
  5. Classical dynamics
  6. Dynamical stabilization condition
  7. Typical space-time scales
  8. Strategy of proof and plan of the paper
  9. Toolbox of semiclassical analysis and reduction of the problem to a resolvent estimate
  10. Weyl quantization and semiclassical defect measures
  11. Weyl--Heisenberg translation operators
  12. Resolvent estimate
  13. A priori conditions of stabilization: Proof of Propositions \ref{['prop:aprioriNC']} and \ref{['prop:aprioriSC']}
  14. A priori necessary condition
  15. A priori sufficient condition
  16. ...and 6 more sections

Key Result

Proposition 1.2

Assume $V$ is subject to eq:assumV and $b \in L^\infty(\mathbf{R}^d)$ is a non-negative damping coefficient. Then if eq:eq is uniformly stable, $b$ satisfies the Uniform Geometric Control Condition where

Figures (1)

  • Figure 1: Black and white indicate the points where $b$ is equal to $1$ or $0$ respectively.

Theorems & Definitions (41)

  • Definition 1.1: Uniform stabilization
  • Proposition 1.2: A priori necessary condition for uniform stability
  • Remark 1.3
  • Proposition 1.4: A priori sufficient condition for uniform stability
  • Remark 1.6
  • Theorem 1.7: Necessary condition for uniform stability
  • Corollary 1.8: Characterization of uniform stability for uniformly continuous damping coefficients
  • Definition 1.9: Stabilization condition --- Dynamical formulation
  • Proposition 1.10
  • Conjecture 1.11
  • ...and 31 more