Sunto
Consideriamo vari sistemi di PDE del primo ordine con dissipazione e leggi di conservazione parziali. Tale classe include modelli del tipo studiato da S. Jin e Z. Xin per i modelli di gas alle discrete velocità, come il sistema di Broadwell. Il modello di S. Jin e Z. Xin ammette una regione convessa, compatta, positivamente invariante, in funzione del sistema all’equilibrio. Di conseguenza è ottenuta l’esistenza di soluzioni deboli globali per il problema di Cauchy per dati grandi. Per sistemi più generali l’esistenza globale di una soluzione ad entropia limitata risulta un’assunzione di base in questo lavoro. I dati sono unidimensionali o spazialmente periodici o inL 2. Proviamo che la soluzione di entropia tende allo stato di equilibrio; quest’ultimo o è zero o è determinato dal valor medio delle componenti conservate. Sottolineiamo che non richiediamo ipotesi sulla nonlinearità del sistema nello stato base di equilibrio. Forniamo due diverse dimostrazioni di stabilità. La prima usa il metodo di compattezza condensata ed è piuttosto efficiente. Ad esempio diversi modelli quasi lineari. Ma tale prova non fornisce una velocità di decadimento. La seconda usa una stima di dispersione per la parte principale del modello. Si applica a dati periodici e richiede una forte ipotesi di semilinearità, ma fornisce un decadimento di tipo esponenziale per la norma diL 2. Auspichiamo che possa estendersi in contesti multidimensionali.
Abstract
We consider various first-order systems of PDEs with partial dissipation, as well as partial conservation. This class includes relaxation models, for instance the one designed by S. Jin and Z. Xin, as well as discrete velocity models for gases, as the Broadwell system. As we showed in a recent paper, the Jin-Xin model admits a convex compact positively invariant region, whenever the equilibrium system does. As a by-product, we obtained the existence of global weak solutions for the Cauchy problem with large data. For more general systems, the global existence of a uniformly bounded entropy solution will be a basic assumption in this work. We consider one-dimensional data which are either space periodic or square integrable. We prove that the (expected globally bounded) entropy solution relaxes to the equilibrium state; the latter is either zero or is determined by the mean value of the conserved components. We emphasize that we do not need any assumption about the nonlinearity of the underlying equilibrium system. We give two different proofs of the stabilization, which apply in different contexts. The first one uses compensated compactness and has a rather broad efficiency. For instance, it applies to several quasi-linear models. But the convergence result does not provide any decay rate in the periodic setting. The other one uses a dispersion estimate for the principal part of the model. It applies to periodic data and needs the strong assumption of semi-linearity, but yields an exponential decay in theL 2-norm. We expect that it could extend to multi-dimensional contexts.
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This research was done from one part in accomplishment of the European IHP project «HYKE», contract #XXX, while visiting the Dipartimento di Matematica of the Università degli Studi di Ferrara, and the CIRAM in the Università di Bologna. The author is happy to thank Professors M. Padula, F. Ancona, A. Corli and T. Ruggeri for their kind invitations and the same persons, plus A. Bressan and L. Tartar for stimulating discussions.
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Serre, D. The stability of constant equilibrium states in relaxation models. Ann. Univ. Ferrara 48, 253–274 (2002). https://doi.org/10.1007/BF02824749
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DOI: https://doi.org/10.1007/BF02824749