Vol. 13, No. 5, 2020

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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
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.

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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