Chapter 7 Recent Results on Hydrodynamic Limits

Abstract

Hydrodynamic limits study the relation between the fluid and kinetic descriptions of a transport phenomenon. In different area of physics and engineering, kinetic models have been developed and studied to shed some light on phenomenon far from thermodynamical equilibrium. Those models are far more accurate. However, they are still numerically too much costly, and the fluid descriptions cannot be completely avoided. It is then crucial to describe precisely the relation between these two kind of models, especially close to discontinuities, where the difference between the models are expected to be the deepest. We present in this chapter a general theory of hydrodynamic limits based on the so-called relative entropy method. We give general conditions ensuring that a given conservation law (or balance law) can be obtained as hydrodynamic limit from a given kinetic equation. Several examples in gas dynamics are provided. Relative entropy methods are known to provide asymptotic limits until the first time of appearance of singularities. We give first hints how to apply it beyond certain kind of singularities.