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

TL;DR

This paper investigates the long- and short-term dynamics of hypocoercive evolution equations using modal decompositions, providing uniform estimates across modes and generalizing previous kinetic Lorentz equation results.

Contribution

It introduces a generalized approach for analyzing hypocoercive PDEs with modal decompositions, extending prior work on the kinetic Lorentz equation.

Findings

Derived uniform estimates for solutions across all modes.

Generalized hypocoercivity analysis to a broader class of evolution equations.

Extended previous results from kinetic Lorentz equations to general generators.

Abstract

The long- and short-time behavior of solutions to dissipative evolution equations is studied by applying the concept of hypocoercivity. Aiming at partial differential equations that allow for a modal decomposition, we compute estimates that are uniform with respect to all modes. While the special example of the kinetic Lorentz equation was treated in previous work of the authors, that analysis is generalized here to general evolution equations having a scaled family of generators.