Very weak solutions to hypoelliptic wave equations

TL;DR

This paper investigates the well-posedness of hypoelliptic wave equations with various regularity coefficients on graded Lie groups, introducing the concept of very weak solutions for distributional coefficients and demonstrating their uniqueness and consistency with classical solutions.

Contribution

It extends the theory of wave equations on graded Lie groups by establishing well-posedness with H"older and distributional coefficients using very weak solutions, a novel approach in this context.

Findings

Well-posedness for H"older coefficients in ultradistribution spaces.

Existence and uniqueness of very weak solutions for distributional coefficients.

Consistency of very weak solutions with classical solutions when they exist.

Abstract

In this paper we study the Cauchy problem for the wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups when the time-dependent non-negative propagation speed is regular, H\"older, and distributional. For H\"older coefficients we derive the well-posedness in the spaces of ultradistributions associated to Rockland operators on graded groups. In the case when the propagation speed is a distribution, we employ the notion of "very weak solutions" to the Cauchy problem, that was already successfully used in similar contexts in [GR15] and [RT17b]. We show that the Cauchy problem for the wave equation with the distributional coefficient has a unique "very weak solution" in an appropriate sense, which coincides with classical or distributional solutions when the latter exist. Examples include the time dependent wave equation for the sub-Laplacian on the Heisenberg group or on general stratified Lie groups, or $p$-evolution equations for higher order operators on $\mathbb{R}^{n}$ or on groups, the results already being new in all these cases.