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Hölder damping for fractional wave equations

Jian Wang, Ruoyu P. T. Wang

TL;DR

This work analyzes energy decay for damped fractional wave equations on closed manifolds under low Hölder regularity damping. It combines semiclassical resolvent analysis with precise commutator bounds for Hölder multipliers to derive a polynomial energy decay $E(u,t)\le C\langle t\rangle^{-\gamma_\#}$, where $\gamma_\#$ and the delay parameter $\nu_\#$ depend explicitly on the damping Hölder exponent $\beta$ and the fractional order $\alpha$: $\gamma_\#=\frac{2}{1-2\left(1+\frac{\nu_\#}{\alpha}\right)}$ and $\nu_\#=\min(-1,\,2\beta+\frac{\alpha}{2}-2)$. The analysis hinges on a semiclassical reduction of the resolvent problem and a two-regime (elliptic and propagation) decomposition, with GCC ensuring energy decay despite low regularity. The results generalize and quantify earlier works by showing how damping regularity governs decay rates, providing a sharp link between low-regularity damping and slower energy dissipation with potential implications for linearized gravity water waves and controlled damping designs.

Abstract

For fractional wave equations with low Hölder regularity damping, we establish quantitative energy decay rates for their solutions when the geometric control condition holds. The energy decay rates depend explicitly on the Hölder regularity of the damping. In particular, we show damping functions with lower Hölder regularities that below a certain threshold give slower energy decay.

Hölder damping for fractional wave equations

TL;DR

This work analyzes energy decay for damped fractional wave equations on closed manifolds under low Hölder regularity damping. It combines semiclassical resolvent analysis with precise commutator bounds for Hölder multipliers to derive a polynomial energy decay , where and the delay parameter depend explicitly on the damping Hölder exponent and the fractional order : and . The analysis hinges on a semiclassical reduction of the resolvent problem and a two-regime (elliptic and propagation) decomposition, with GCC ensuring energy decay despite low regularity. The results generalize and quantify earlier works by showing how damping regularity governs decay rates, providing a sharp link between low-regularity damping and slower energy dissipation with potential implications for linearized gravity water waves and controlled damping designs.

Abstract

For fractional wave equations with low Hölder regularity damping, we establish quantitative energy decay rates for their solutions when the geometric control condition holds. The energy decay rates depend explicitly on the Hölder regularity of the damping. In particular, we show damping functions with lower Hölder regularities that below a certain threshold give slower energy decay.
Paper Structure (7 sections, 4 theorems, 52 equations, 1 figure)

This paper contains 7 sections, 4 theorems, 52 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Commutators with Hölder functions
  3. Semiclassical calculus
  4. Commutator with Hölder functions
  5. Resolvent estimates and Proof of Theorem
  6. Elliptic regime
  7. Propagation regime

Key Result

Lemma 2.1

There exists $C>0$ such that for all $f\in C^{0,\beta}(M)$ with $\beta\in [0,1]$ and $a\in C^{\infty}_c(T^*M)$, for all $h>0$

Figures (1)

  • Figure 1: Numerical illustration of eigenfunctions for $P(\lambda)$ in § \ref{['sec:proof']} for damping with different regularities on the circle $\mathbb T=\mathbb{R}/2\pi \mathbb Z$. Blue curves are the real parts, imaginary parts, and absolute values of the eigenfunctions. Red dashed curves are the damping functions. (A) Damping $\chi = \mathbbm 1_{(-\frac{\pi}{2},\frac{\pi}{2})}$. Eigenvalue $\lambda\sim 13.03-0.03i$. (B) Damping $\chi(x) = 1+\frac{1}{2}(\tanh(20(x-\frac{\pi}{2}))-\tanh(20(x+\frac{\pi}{2})))$. Eigenvalue $\lambda\sim 13.01-0.14i$.

Theorems & Definitions (10)

  • Definition 1.1
  • Lemma 2.1
  • proof
  • Proposition 3.1: Resolvent estimate for $C^{0,\beta}$-damping
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • proof : Proof of Proposition \ref{['prop:resolvent']}
  • proof : Proof of Theorem