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The Quintic Wave Equation with Kelvin-Voigt Damping: Strichartz estimates, Well-posedness and Global Stabilization

Marcelo Moreira Cavalcanti, Valeria Neves Domingos Cavalcanti

Abstract

This paper investigates the critical quintic wave equation in a 3D bounded domain subject to locally distributed Kelvin-Voigt damping. The study tackles two major mathematical challenges: the severe loss of derivatives induced by the localized thermo-viscous dissipation and the aggressive nature of the critical nonlinear term. First, we establish a robust well-posedness theory for arbitrarily large initial data. By shifting the analysis to the frequency space via a Littlewood-Paley decomposition , we employ Bernstein's inequalities to lift the damping term into an $L^2$ framework, allowing Strichartz estimates to be applied flawlessly. In the second part, we prove the uniform exponential stabilization of the energy. To overcome the reduction of the residual to the $H^{-2}$ level caused by the Kelvin-Voigt mechanism, we utilize the microlocal defect measure framework. The core of our stabilization proof relies on combining the critical Strichartz regularity $L^4_t L^{12}_x$ with a sharp Unique Continuation Property (UCP) to close the observability argument. Furthermore, we demonstrate that this microlocal mechanism is perfectly compatible with non-invasive damping geometries of arbitrarily small Lebesgue measure, successfully circumventing the geometric obstruction of trapped rays.

The Quintic Wave Equation with Kelvin-Voigt Damping: Strichartz estimates, Well-posedness and Global Stabilization

Abstract

This paper investigates the critical quintic wave equation in a 3D bounded domain subject to locally distributed Kelvin-Voigt damping. The study tackles two major mathematical challenges: the severe loss of derivatives induced by the localized thermo-viscous dissipation and the aggressive nature of the critical nonlinear term. First, we establish a robust well-posedness theory for arbitrarily large initial data. By shifting the analysis to the frequency space via a Littlewood-Paley decomposition , we employ Bernstein's inequalities to lift the damping term into an framework, allowing Strichartz estimates to be applied flawlessly. In the second part, we prove the uniform exponential stabilization of the energy. To overcome the reduction of the residual to the level caused by the Kelvin-Voigt mechanism, we utilize the microlocal defect measure framework. The core of our stabilization proof relies on combining the critical Strichartz regularity with a sharp Unique Continuation Property (UCP) to close the observability argument. Furthermore, we demonstrate that this microlocal mechanism is perfectly compatible with non-invasive damping geometries of arbitrarily small Lebesgue measure, successfully circumventing the geometric obstruction of trapped rays.
Paper Structure (33 sections, 12 theorems, 117 equations)

This paper contains 33 sections, 12 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. The phenomenon of energy concentration
  3. The drawback of Galerkin's method
  4. Literature Overview
  5. Contributions of the present article
  6. Global Weak Solutions via $C^1$ Truncation
  7. Step 1: The $C^1$ Truncated Problem
  8. Step 2: Galerkin Method for the Truncated Problem
  9. Step 3: The Critical Limit $k \to \infty$
  10. Local Well-posedness of Strong Solutions via Galerkin Method and Small Data
  11. Uniqueness and Weak-Strong Uniqueness Principles
  12. Uniqueness of Strong Solutions
  13. Uniqueness in the Strichartz Class
  14. Littlewood-Paley Strategy
  15. The Philosophy: Frequencies Instead of Space
  16. ...and 18 more sections

Key Result

Lemma 2.1

For each $k \in \mathbb{N}$, the following properties hold:

Theorems & Definitions (24)

  • Remark 1.1
  • Lemma 2.1: Properties of $f_k$
  • proof
  • Theorem 2.2: Existence of Global Weak Solutions
  • Theorem 3.1: Local Well-posedness of Strong Solutions
  • Theorem 4.1: Uniqueness of Strong Solutions
  • proof
  • Theorem 4.2: Uniqueness in the Strichartz Class
  • proof
  • Lemma 5.1: Uniform integrability and weak compactness
  • ...and 14 more