2023/09/22 by Matthieu Léautaud, Léautaud, Matthieu
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods for differential equations #Spectral Theory (math.SP) #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2309.12709
openalex publication_date 2023/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02
We consider the damped wave equation on a compact manifold. We propose different ways of measuring decay of the energy (time averages of lower energy levels, decay for frequency localized data...) and exhibit links with resolvent estimates on the imaginary axis. As an application we prove a universal logarithmic lower resolvent bound on the imaginary axis for the damped wave operator when the Geometric Control Condition (GCC) is not satisfied. This is to be compared to the uniform boundedness of the resolvent on that set when GCC holds. The proofs rely on (i) various (re-)formulations of the damped wave equation as a conservative hyperbolic part perturbed by a lower order damping term;(ii) a "Plancherel-in-time" argument as in classical proofs of the Gearhart-Huang-Prüss theorem; and (iii) an idea of Bony-Burq-Ramond of propagating a coherent state along an undamped trajectory up to Ehrenfest time.