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Geometry of wave damping on the torus

Kiril Datchev, Perry Kleinhenz, Antoine Prouff

TL;DR

The paper develops a geometric framework for the damped wave equation on $\mathbb{T}^2$ in which the energy decay rate is controlled by the local behavior of the damping near glancing regions. By introducing the order $\eta$ of glancing points and an averaging operator $A_v(W)$ along glancing directions, the authors reduce the 2D resolvent problem to 1D averaged problems and derive explicit polynomial decay rates $E(u,t) \lesssim t^{-\alpha}$ with $\alpha$ determined by $\beta$ and $\eta$. They prove a general theorem linking 2D decay to averaged 1D bounds, and establish that damping geometries achieving improved rates are generic for both polygonal and $C^2$ dampings, with explicit improvements $\alpha = 1 - \frac{1}{\beta+1+3}$ for polygons and $\alpha = 1 - \frac{1}{\beta+\frac{1}{2}+3}$ for smoothly curved boundaries. Altogether, the work generalizes prior results and shows that sharper local boundary geometry and damping growth near glancing points yield stronger stabilization, with broad implications for design of damping regions in toroidal domains.

Abstract

Energy decay rates of damped waves on the torus depend on the behavior of the damping near the undamped region and on the geometry of the damped set. In this paper we refine these geometric considerations, by introducing the concept of order of a glancing undamped point, and estimating decay rates in terms of this order. The proof is based on generalizing an averaging argument due to Sun. We also show that damping sets which attain these improvements are generic among polygons and smooth curves.

Geometry of wave damping on the torus

TL;DR

The paper develops a geometric framework for the damped wave equation on in which the energy decay rate is controlled by the local behavior of the damping near glancing regions. By introducing the order of glancing points and an averaging operator along glancing directions, the authors reduce the 2D resolvent problem to 1D averaged problems and derive explicit polynomial decay rates with determined by and . They prove a general theorem linking 2D decay to averaged 1D bounds, and establish that damping geometries achieving improved rates are generic for both polygonal and dampings, with explicit improvements for polygons and for smoothly curved boundaries. Altogether, the work generalizes prior results and shows that sharper local boundary geometry and damping growth near glancing points yield stronger stabilization, with broad implications for design of damping regions in toroidal domains.

Abstract

Energy decay rates of damped waves on the torus depend on the behavior of the damping near the undamped region and on the geometry of the damped set. In this paper we refine these geometric considerations, by introducing the concept of order of a glancing undamped point, and estimating decay rates in terms of this order. The proof is based on generalizing an averaging argument due to Sun. We also show that damping sets which attain these improvements are generic among polygons and smooth curves.

Paper Structure

This paper contains 12 sections, 15 theorems, 125 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Definitions, notation and main results
  4. Genericity results
  5. Polygonal damping sets
  6. $C^2$ damping sets
  7. Reduction to Averaging
  8. Proof of 1d combination results, Proposition \ref{['prop:1dresolve']}
  9. Proof of Proposition \ref{['prop:order']}
  10. Proofs of results for polygonal damping sets
  11. Proofs of results for $C^2$ damping sets
  12. Appendix

Key Result

Theorem 1.1

Consider the damped wave equation where $W \in W^{9, \infty}(\mathbb{T}^2)$ is nonnegative, not identically zero, and $|\partial^\gamma W| \lesssim W^{1-\frac{|\gamma|}{4}}$ for $|\gamma| \le 2$. Suppose that the damping set is as in Figure f:1ex, and let $d(z)=\text{dist}(z, \mathbb{T}^2 \backslash \omega)$ be the distance from $z$ to the undamped region. Assume there exists $\beta\ge9$ such th

Figures (9)

  • Figure 1: Three examples of damping sets $\omega$ on the torus, bounded by (A) a cylinder, (B) a curve which at the six blue points has nonzero curvature, and (C) a polygon. Blue indicates points where undamped geodesics (shown in gray) touch $\partial \omega$; in (C) the red geodesic intersects $\omega$ so it is not undamped.
  • Figure 2: The disk of diameter $1/\sqrt2$ has eight glancing directions (one for each eighth root of unity) and six glancing lines. The horizontal and vertical glancing lines are one-sided, and the diagonal ones are two-sided. If the damping set is a superset of this disk, then its glancing directions are a subset of these eight. If the damping set is a suitable subset of this disk, then the glancing directions are still the same; see Figure \ref{['f:disk2']} for examples.
  • Figure 3: The first two are one-sided glancing points, obeying \ref{['e:order-1-sided']}. The last two are two-sided glancing points, obeying respectively the first and the second of \ref{['e:order-2-sided']}.
  • Figure 4: A point of order 1. The gray is $0 \le C_{in}^{-1}y \le x \le C_{in} y$, and the black is $|y|=C_{out}x$ with the values $C_{in}=1.05$, $C_{out}=3$, chosen far from optimal for emphasis. In the left hand side, $t$ is the coordinate along $v$ and $s$ is the coordinate along $v^{\bot}$.
  • Figure 5: With respect to the horizontal direction, a point of order $1/2$ (damping region $\frac{1}{2}x^{1/2} < y < 2x^{1/2}$) and a point of order $1/4$ (damping region $\frac{1}{2} x^{1/4} < y < 2 x^{1/4}$). With respect to the vertical direction, the orders are $2$ and $4$.
  • ...and 4 more figures

Theorems & Definitions (45)

  • Theorem 1.1
  • Remark 1.2
  • Definition 1.3: Glancing lines
  • Lemma 1.4
  • Definition 1.5: Glancing points
  • Remark 1.6
  • Definition 1.7
  • Remark 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 35 more