In this paper, we start a general study on relaxation hyperbolic systems which violate the Shizuta–Kawashima coupling condition ([SK]). This investigation 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 role 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 systems 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.
On Relaxation Hyperbolic Systems Violating the Shizuta-Kawashima Condition / Mascia, Corrado; Natalini, Roberto. - In: ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS. - ISSN 0003-9527. - STAMPA. - 195:(2010), pp. 729-762. [10.1007/s00205-009-0225-x]
On Relaxation Hyperbolic Systems Violating the Shizuta-Kawashima Condition
MASCIA, Corrado;NATALINI, Roberto
2010
Abstract
In this paper, we start a general study on relaxation hyperbolic systems which violate the Shizuta–Kawashima coupling condition ([SK]). This investigation 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 role 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 systems 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.