Diffusion phenomena for partially dissipative hyperbolic systems

TL;DR

This paper analyzes the diffusive behavior of partially dissipative hyperbolic systems with time-dependent coefficients, showing they asymptotically resemble solutions of parabolic equations under certain conditions.

Contribution

It provides precise estimates and a novel combination of elliptic WKB analysis and exponential stability techniques to understand the diffusive structure of these systems.

Findings

Solutions are asymptotically equivalent to parabolic equations

Established diffusive estimates for systems with time-dependent coefficients

Combined WKB analysis with stability results for comprehensive understanding

Abstract

In this note we provide some precise estimates explaining the diffusive structure of partially dissipative systems with time-dependent coefficients satisfying a uniform Kalman rank condition. Precisely, we show that under certain (natural) conditions solutions to a partially dissipative hyperbolic system are asymptotically equivalent to solutions of a corresponding parabolic equation. The approach is based on an elliptic WKB analysis for small frequencies in combination with exponential stability for large frequencies due to results of Beauchard and Zuazua and arguments of perturbation theory.