Natalini R.; On relaxation hyperbolic systems violating the Shizuta–Kawashima condition

Abstract. In this paper, we start a general study on relaxation hyperbolic sys-tems which violate the Shizuta–Kawashima coupling condition ([SK]). This in-vestigation is motivated by the fact that this condition is in general not satisfied by various physical systems, and almost all the time in several space dimensions. First, we explore the rôle of entropy functionals around equilibrium solutions, which may be not constant, proposing a stability condition for such solutions. Then we find strictly dissipative entropy functions for one dimensional 2 × 2 sys-tems which violate [SK] condition. Finally, we prove the existence of global smooth solutions for a class of systems such that condition [SK] does not hold, but which are linearly degenerated in the non dissipative directions. 1