Logarithmic decay for damped hypoelliptic wave and Schr{\"o}dinger equations
Abstract
We consider damped wave (resp. Schr{\"o}dinger and plate) equations driven by a hypoelliptic "sum of squares" operator L on a compact manifold and a damping function b(x). We assume the Chow-Rashevski-H{\"o}rmander condition at rank k (at most k Lie brackets needed to span the tangent space) together with analyticity of M and the coefficients of L. We prove decay of the energy at rate (resp. ) for data in the domain of the generator of the associated group. We show that this decay is optimal on a family of Grushin-type operators. This result follows from a perturbative argument (of independent interest) showing, in a general abstract setting, that quantitative approximate observability/controllability results for wave-type equations imply a priori decay rates for associated damped wave, Schr{\"o}dinger and plate equations. The adapted quantitative approximate observability/controllability theorem for hypoelliptic waves is obtained by the authors in [LL19, LL17].
Keywords
Cite
@article{arxiv.2006.05122,
title = {Logarithmic decay for damped hypoelliptic wave and Schr{\"o}dinger equations},
author = {Camille Laurent and Matthieu Léautaud},
journal= {arXiv preprint arXiv:2006.05122},
year = {2020}
}