Stabilisation of wave equations on the torus with rough dampings

Abstract

For the damped wave equation on a compact manifold with {\em continuous} dampings, the geometric control condition is necessary and sufficient for {uniform} stabilisation. In this article, on the two dimensional torus, in the special case where $a(x)=\sum \_{j=1}^{N}a\_j1\_x\in R\_j$ ($R\_j$ are polygons), we give a very simple necessary and sufficient geometric condition for uniform stabilisation. We also propose a natural generalization of the geometric control condition which makes sense for $L^{\infty }$ dampings. We show that this condition is always necessary for uniform stabilisation (for any compact (smooth) manifold and any $L^{\infty }$ damping), and we prove that it is sufficient in our particular case on ${\mathbb{T}}^2$ (and for our particular dampings).

References in corpus (5)

• Decay rates for the damped wave equation on the torus
• High-frequency propagation for the Schroedinger equation on the torus
• Refined and microlocal Kakeya-Nikodym bounds for eigenfunctions in two dimensions
• Stabilization of Damped Waves on Spheres and Zoll Surfaces of Revolution
• Bilinear Kakeya-Nikodym averages of eigenfunctions on compact Riemannian surfaces

Cited by in corpus (2)

• Observability of the Schr{ö}dinger equation with subquadratic confining potential in the Euclidean space
• Sharp resolvent estimate for the Baouendi-Grushin operator and applications