Decay rates for the damped wave equation on the torus

TL;DR

This paper investigates the decay rates of energy in the damped wave equation on the torus when damping does not satisfy the Geometric Control Condition, linking controllability of Schrödinger equations to decay rates and analyzing the influence of damping regularity.

Contribution

It establishes a connection between Schrödinger controllability and wave decay rates, and provides precise decay estimates depending on damping regularity and geometry.

Findings

Decay rate is at most 1/t when GCC is not satisfied.

Smooth damping yields decay close to 1/t, with a small epsilon loss.

Discontinuous damping can limit decay to no faster than 1/t^{2/3}.

Abstract

We address the decay rates of the energy for the damped wave equation when the damping coefficient $b$ does not satisfy the Geometric Control Condition (GCC). First, we give a link with the controllability of the associated Schr\"odinger equation. We prove in an abstract setting that the observability of the Schr\"odinger group implies that the semigroup associated to the damped wave equation decays at rate $1/\sqrt{t}$ (which is a stronger rate than the general logarithmic one predicted by the Lebeau Theorem). Second, we focus on the 2-dimensional torus. We prove that the best decay one can expect is $1/t$, as soon as the damping region does not satisfy GCC. Conversely, for smooth damping coefficients $b$, we show that the semigroup decays at rate $1/t^{1-\eps}$, for all $\eps >0$. The proof relies on a second microlocalization around trapped directions, and resolvent estimates. In the case where the damping coefficient is a characteristic function of a strip (hence discontinuous), St\'{e}phane Nonnenmacher computes in an appendix part of the spectrum of the associated damped wave operator, proving that the semigroup cannot decay faster than $1/t^{2/3}$. In particular, our study shows that the decay rate highly depends on the way $b$ vanishes.