On Lipschitz vector fields and the Cauchy problem in homogeneous groups

TL;DR

This paper introduces Lipschitz horizontal vector fields in homogeneous groups, explores their properties, and analyzes the well-posedness of the associated Cauchy problem, highlighting differences from Euclidean cases.

Contribution

It defines a new class of Lipschitz vector fields in homogeneous groups and investigates their Cauchy problem, including uniqueness and counterexamples to general well-posedness.

Findings

Established equivalences for Lipschitz horizontal vector fields

Proved uniqueness of solutions at equilibrium points

Provided a counterexample to general well-posedness in this setting

Abstract

We introduce a class of "Lipschitz horizontal" vector fields in homogeneous groups, for which we show equivalent descriptions, e.g. in terms of suitable derivations. We then investigate the associated Cauchy problem, providing a uniqueness result both at equilibrium points and for vector fields of an involutive submodule of Lipschitz horizontal vector fields. We finally exhibit a counterexample to the general well-posedness theory for Lipschitz horizontal vector fields, in contrast with the Euclidean theory.