Connection of hypocoercivity and hypocontractivity via the Cayley transform

TL;DR

This paper explores the relationship between hypocoercivity and hypocontractivity in evolution equations, providing new proofs for decay characterization and analyzing the impact of different representations on numerical error estimates.

Contribution

It introduces new proofs for short-time decay characterization and examines how different representations affect numerical error estimates in hypocoercive and hypocontractive systems.

Findings

New proofs for solution decay characterization

Maximally coercive/contractive system representations

Impact of representations on numerical error estimates

Abstract

The concepts of hypocoercivity and hypocontractivity and their relationship are studied for semi-dissipative continuous-time and discrete-time evolution equations in a Hilbert space setting. New proofs for the characterization of the short-time decay of the solution from the initial value are presented, that in particular characterize the constants in the leading terms of the solution when expanded in time. Maximally coercive/contractive representations of hypocoercive and hypocontractive semi-dissipative systems are presented, as well as the effect of different representations on the error estimates for the numerical solution.