The stability of constant equilibrium states in relaxation models

Abstract

We consider various first-order systems of PDEs with partial dissipation, as well as partial conservation. This class includes relaxation models, for instance the one designed by S. Jin and Z. Xin, as well as discrete velocity models for gases, as the Broadwell system. As we showed in a recent paper, the Jin-Xin model admits a convex compact positively invariant region, whenever the equilibrium system does. As a by-product, we obtained the existence of global weak solutions for the Cauchy problem with large data. For more general systems, the global existence of a uniformly bounded entropy solution will be a basic assumption in this work. We consider one-dimensional data which are either space periodic or square integrable. We prove that the (expected globally bounded) entropy solution relaxes to the equilibrium state; the latter is either zero or is determined by the mean value of the conserved components. We emphasize that we do not need any assumption about the nonlinearity of the underlying equilibrium system. We give two different proofs of the stabilization, which apply in different contexts. The first one uses compensated compactness and has a rather broad efficiency. For instance, it applies to several quasi-linear models. But the convergence result does not provide any decay rate in the periodic setting. The other one uses a dispersion estimate for the principal part of the model. It applies to periodic data and needs the strong assumption of semi-linearity, but yields an exponential decay in theL2-norm. We expect that it could extend to multi-dimensional contexts.