Fourier-Cattaneo equation: stochastic origin, variational formulation, and asymptotic limits

TL;DR

This paper develops a variational framework for the Fourier-Cattaneo system, linking it to large-deviation principles, and analyzes its asymptotic diffusive and hyperbolic limits.

Contribution

It introduces a novel variational formulation for the FC system inspired by large deviations, enabling new solution concepts and limit analysis.

Findings

Established a variational structure for the FC system.

Proved an a priori estimate connecting the structure to Lyapunov and Fisher information.

Analyzed the diffusive and hyperbolic asymptotic limits of the system.

Abstract

We introduce a variational structure for the Fourier-Cattaneo (FC) system which is a second-order hyperbolic system. This variational structure is inspired by the large-deviation rate functional for the Kac process which is closely linked to the FC system. Using this variational formulation we introduce appropriate solution concepts for the FC equation and prove an a priori estimate which connects this variational structure to an appropriate Lyapunov function and Fisher information, the so-called FIR inequality. Finally, we use this formulation and estimate to study the diffusive and hyperbolic limits for the FC system.