Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy

We study the asymptotic time behavior of global smooth solutions to general entropy dissipative hyperbolic systems of balance law in m space dimensions, under the Shizuta-Kawashima condition. We show that these solutions approach constant equilibrium state in the Lp-norm at a rate O(t^(-m/2(1-1/p))), as t tends to $\infty$, for p in [min (m,2),+ \infty]. Moreover, we can show that we can approximate, with a faster order of convergence, theconservative part of the solution in terms of the linearized hyperbolic operator for m >= 2, and by a parabolic equation in the spirit of Chapman-Enskog expansion. The main tool is given by a detailed analysis of the Green function for the linearized problem

Relaxation approximation of the Kerr Model for the three dimensional initial-boundary value problem

The electromagnetic waves propagation in a non linear medium can be described by the Kerr model in case of instantaneous response of the material, or by the Kerr-Debye model if the material exhibits a finite response time. Both models are quasilinear hyperbolic and are endowed with a dissipative entropy. Initial-boundary value problem with the maximal dissipative impedance boundary condition is considered. When the response time is fixed, in the one dimensional and the two dimensional transverse electric cases, the global existence of smooth solutions for the Kerr-Debye system is established. When the response time tends to zero, the convergence of the Kerr-Debye model to the Kerr model is proved in the general case: the Kerr model is the zero relaxation limit of the Kerr-Debye mode

Relaxation Approximation of some Initial-Boundary Value Problem for p-Systems

We consider the Suliciu model which is a relaxation approximation of the $p$-system. In the case of the Dirichlet boundary condition we prove that the local smooth solution of the $p$-system is the zero limit of the Suliciu model solutions