Relative entropy methods and relaxation limits

We shall analyze relative entropy methods in connections with singular limits, both for the hyperbolic-to-hyperbolic and for the hyperbolic-to-parabolic relaxations. For the former case, we recall in particular the results proved in [Lattanzio, Tzavaras, ARMA 2006] for the hyperbolic stress relaxation limit for multidimensional elasticity. Moreover, for diffusive relaxations, we obtain a relative entropy estimate in the general case of a large friction theory converging toward gradient flow equations. The framework considered applies in particular to the Euler equations with large friction converging toward the porous medium equation [Lattanzio, Tzavaras, SIMA 2013], and to the diffusive relaxation limits for Keller-Segel type models and for Euler-Korteweg equations with large friction.