Global Existence of Smooth Solutions for Partially Dissipative Hyperbolic Systems with a Convex Entropy

The strong relaxation limit of the multidimensional isothermal Euler equations

“…For fixed τ > 0, the existence of global smooth solutions to (1) follows from a result by Yong [10] (which is the analogue of [2] in the multidimensional framework). We refer the reader to [8] for the existence of global smooth solutions in the isentropic case.…”

Entropy Principle and Recent Results in Non-Equilibrium Theories

“…In fact, not only every nesting theories of RET are governed by symmetric hyperbolic systems with the property of well posedness of local (in time) Cauchy problem (see Section 5) but there can exists also global smooth solutions due to the overlap between the first 5 conservation laws and the remaining dissipative ones. In fact, for generic hyperbolic systems of balance laws (121), endowed with a convex entropy law, and dissipative, the so called Kawashima-Shizuta K-condition [110] becomes a sufficient condition for the existence of global smooth solutions, provided the initial data are sufficiently smooth (Hanouzet and Natalini [111], Wen-An Yong [112], Bianchini, Hanouzet and Natalini [113], see also the Dafermos book [114]). Theorem[Global Existence].…”

Decay property of regularity-loss type for the Euler-Maxwell system

“…We can check that our system verifies the stability condition formulated above; for the details, we refer the reader to [23]. However, the dissipation matrix L is not real symmetric and therefore we can not apply the general theory on the dissipative structure developed in [20,22,10,6,24,11,1,12] to our system. This situation is quite similar to that for the dissipative Timoshenko system.…”

“…Consequently, our system (2.2) is not included in a class of systems considered in [1,6,10,11,12,20,22,24].…”

“…We refer the reader to [6,24,1,12]. All these works were, however, based on the standard dissipative structure formulated in [20,22,10] and hence are different from the present paper.…”