Subelliptic Wave Equations with Log-Lipschitz coefficients

TL;DR

This paper investigates the well-posedness of subelliptic wave equations with Log-Lipschitz coefficients on Lie groups, demonstrating finite regularity loss and extending classical results to more general settings.

Contribution

It establishes well-posedness for subelliptic wave equations with Log-Lipschitz coefficients on Lie groups, showing finite regularity loss unlike H"older cases, and extends results to Hilbert spaces.

Findings

Proves well-posedness with finite regularity loss

Extends classical wave equation results to subelliptic and Lie group contexts

Demonstrates differences between Log-Lipschitz and H"older coefficients

Abstract

In this paper we study the Cauchy problem for the wave equations for sums of squares of left invariant vector fields on compact Lie groups and also for hypoelliptic homogeneous left-invariant differential operators on graded Lie groups (the positive Rockland operators), when the time-dependent propagation speed satisfies a Log-Lipschitz condition. We prove the well-posedness in the associated Sobolev spaces exhibiting a finite loss of regularity with respect to the initial data, which is not true when the propagation speed is a ${\rm H\ddot{o}lder}$ function. We also indicate an extension to general Hilbert spaces. In the special case of the Laplacian on $\mathbb R^n$, the results boil down to the celebrated result of Colombini-De Giorgi and Spagnolo.