Weak solutions for equations defined by accretive operators II: relaxation limits

Hydrodynamic limits for kinetic equations and the diffusive approximation of radiative transport for acoustic waves

“…Our analysis encompasses various results on diffusive limits of simpler models [16,5,26] in the L 1 -contraction framework, and extends the diffusive limit analysis to general contractive models with one conservation law. As a technique it treats the hyperbolic and diffusive scales in a common framework, and uses in an essential way the contraction structure but with very little information from the limit equation.…”

“…A number of previously studied models fit into the above framework, including relaxation approximations [13,21], kinetic BGK models [24,4], and the discrete kinetic models in [19,5,26]. Most existing works concern BGK-type collision operators, and our objective is to put these works in a common framework and to develop a theory for general collision operators.…”

“…A novel feature of this work is the analytical method of proof, which is based on the concept of dissipative solutions of [25] and the perturbed test function method ( [9], [26]), and renders the proofs particularly simple and capable of dealing with complex models. Interestingly, the usual Hilbert expansion used for identifying the limiting behavior is now carried to the test functions, and the asymptotic analysis process is particularly appealing.…”

“…In section 3, we give a simplified convergence proof using the setting of dissipative solutions [25], [26]. Contractive kinetic equations provide a general framework for the extension of Kruzhkov theory to kinetic models.…”

“…In addition, as is proved in [25], the notion of dissipative solution is equivalent to the usual notion of Kruzhkov entropy solution, which is familiar from the theory of scalar conservation laws. Dissipative solutions provide a particularly good framework to study relaxation limits (see [26] and the following sections) by using the perturbed test function method of Evans [9]. See [23] for an analogous notion of "accretive solution" for degenerate diffusion equations, and its relations with the entropy solution in [8] (see also the remark following Proposition 11).…”