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

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