Spectral theory of pseudodifferential operators of degree 0 and an application to forced linear waves

Colin de Verdière, Yves

doi:10.2140/apde.2020.13.1521

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Spectral theory of pseudodifferential operators of degree 0 and an application to forced linear waves

Yves Colin de Verdière

Vol. 13 (2020), No. 5, 1521–1537

DOI: 10.2140/apde.2020.13.1521

Abstract

We extend the results of our paper “Attractors for two-dimensional waves with homogeneous Hamiltonians of degree 0,” written with Laure Saint-Raymond, to the case of forced linear wave equations in any dimension. We prove that, in dimension 2, if the foliation on the boundary at infinity of the energy shell is Morse–Smale, we can apply Mourre’s theory and hence get the asymptotics of the forced solution. We also characterize the wavefront sets of the limit Schwartz distribution using radial propagation estimates.

Keywords

forced waves, internal waves, inertial waves, attractors, Mourre theory, spectral theory, limiting absorption principle, escape functions, pseudodifferential operator, Morse–Smale property

Mathematical Subject Classification 2010

Primary: 35B34, 35Q30, 35Q35, 58J40, 76B55

Secondary: 76B70

Milestones

Received: 9 May 2018

Revised: 6 April 2019

Accepted: 12 May 2019

Published: 27 July 2020

Authors

Yves Colin de Verdière
Université Grenoble Alpes Institut Fourier Unité mixte de recherche CNRS-UGA 5582 Saint Martin d’Hères France

Vol. 13, No. 5, 2020