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On Relaxation Hyperbolic Systems Violating the Shizuta–Kawashima Condition

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Abstract

In this paper, we start a general study on relaxation hyperbolic systems which violate the Shizuta–Kawashima ([SK]) coupling condition. This investigation is motivated by the fact that this condition is 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 not be constant, proposing a stability condition for such solutions. Then we find strictly dissipative entropy functions for one dimensional 2 × 2 systems which violate the [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.

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Authors and Affiliations

  1. Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, Piazzale A. Moro, 2, 00185, Rome, Italy

    Corrado Mascia

  2. Istituto per le Applicazioni del Calcolo “M. Picone”, Consiglio Nazionale delle Ricerche, c/o Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 1, 00133, Rome, Italy

    Roberto Natalini

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  1. Corrado Mascia
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  2. Roberto Natalini
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Correspondence to Corrado Mascia.

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Communicated by C. M. Dafermos

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Mascia, C., Natalini, R. On Relaxation Hyperbolic Systems Violating the Shizuta–Kawashima Condition. Arch Rational Mech Anal 195, 729–762 (2010). https://doi.org/10.1007/s00205-009-0225-x

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