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

Stefano Bianchini (1) , Bernard Hanouzet (2) , Roberto Natalini (1)

1 IAC - istituto per le applicazioni del calculo
2 IMB - Institut de Mathématiques de Bordeaux

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.

Domaines: Equations aux dérivées partielles [math.AP]

Stefano Bianchini, Bernard Hanouzet, Roberto Natalini. Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Communications on Pure and Applied Mathematics, 2007, 60 (11), pp.1559-1622. ⟨hal-00281551⟩