Wave equation for sums of squares on compact Lie groups

Claudia Garetto,Michael Ruzhansky

doi:10.1016/j.jde.2015.01.034

Claudia Garetto, Michael Ruzhansky

Open Access

https://doi.org/10.1016/j.jde.2015.01.034

Copy DOI

Abstract

In this paper we investigate the well-posedness of the Cauchy problem for the wave equation for sums of squares of vector fields on compact Lie groups. We obtain the loss of regularity for solutions to the Cauchy problem in local Sobolev spaces depending on the order to which the Hörmander condition is satisfied, but no loss in globally defined spaces. We also establish Gevrey well-posedness for equations with irregular coefficients and/or multiple characteristics. As in the Sobolev spaces, if formulated in local coordinates, we observe well-posedness with the loss of local Gevrey order depending on the order to which the Hörmander condition is satisfied.

Highlights

In this paper we investigate the well-posedness of the Cauchy problem for time-dependent wave equations associated to sums of squares of invariant vector fields on compact Lie groups
An often encountered example of subelliptic behaviour is a sum of squares of vector fields, extensively analysed by Hörmander [17,18], Oleinik and Radkevich [27], Rothschild and Stein [28], and by many others
The equations are determined by the entries of the matrix symbol of L which we study for this purpose, in particular establishing lower bounds for its eigenvalues in terms of the order to which the Hörmander condition is satisfied

Summary

Introduction

In this paper we investigate the well-posedness of the Cauchy problem for time-dependent wave equations associated to sums of squares of invariant vector fields on compact Lie groups. For invariant operators on compact Lie groups, the sum of squares becomes formally self-adjoint, making the corresponding wave equation hyperbolic, a necessary condition for the analysis of the corresponding Cauchy problem. Already in this setting, we discover a new phenomenon of the loss of the local Gevrey regularity for its solutions. The authors would like to thank Véronique Fischer for stimulating discussions and Ferruccio Colombini for comments

Main results

Fourier analysis and symbolic properties of sub-Laplacians

Reduction to first order system and energy estimates