# Decays for Kelvin-Voigt damped wave equations I : the black box perturbative method

## TL;DR

This paper introduces a perturbative and black box approach to derive decay rates for Kelvin-Voigt damped wave equations using resolvent estimates, achieving optimal logarithmic decay in general cases and specific decay rates in certain geometries.

## Contribution

It combines perturbative methods with black box strategies to establish decay rates for Kelvin-Voigt damped wave equations from standard resolvent estimates, providing a novel analytical framework.

## Key findings

- Achieves logarithmic decay rates, which are optimal in general.
- Demonstrates decay rates in specific geometries like tori.
- Shows the approach's limitations in obtaining optimal results under additional geometric assumptions.

## Abstract

We show in this article how perturbative approaches~from our work with Hitrik (see also the work by Anantharaman-Macia) and the {\em black box} strategy from~ our work with Zworski allow to obtain decay rates for Kelvin-Voigt damped wave equations from quite standard resolvent estimates : Carleman estimates or geometric control estimates for Helmoltz equationCarleman or other resolvent estimates for the Helmoltz equation. Though in this context of Kelvin Voigt damping, such approach is unlikely to allow for the optimal results when additional geometric assumptions are considered (see \cite{BuCh, Bu19}), it turns out that using this method, we can obtain the usual logarithmic decay which is optimal in general cases. We also present some applications of this approach giving decay rates in some particular geometries (tori).