Long- and short-time behavior of hypocoercive evolution equations with higher index via modal decompositions

TL;DR

This paper extends hypocoercivity analysis to unbounded generators with higher index, providing new long- and short-time estimates and an explicit Lyapunov functional, with applications to port-Hamiltonian systems.

Contribution

It generalizes hypocoercivity results to higher index unbounded generators using modal decompositions, which was previously limited to index one.

Findings

Established long-time exponential decay estimates.

Derived short-time asymptotics for higher index generators.

Constructed an explicit Lyapunov functional for stability analysis.

Abstract

Hypocoercivity emerged in kinetic transport theory, allowing to derive exponential long-time estimates for evolution equations. Recently, the short-time asymptotics for equations with dissipative generators were obtained using the hypocoercivity index that is in finite dimensions surprisingly given by a Kalman-type rank condition well-known in control theory. However, the situation for unbounded generators is only understood for index one if modal decompositions are available. Here, we prove long- and short-time estimates for unbounded generators with higher index admitting a modal decomposition. Additionally, an explicit Lyapunov functional is constructed. The result is applied to a class of port-Hamiltonian systems with distributed dissipation.