Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy

Communications on Pure and Applied Mathematics - Tập 60 Số 11 - Trang 1559-1622 - 2007
Stefano Bianchini1, Bernard Hanouzet2, Roberto Natalini3
1Scuola Internazionale Superiore di Studi Avanzati, via Beirut 2–4, 34014 Trieste Italy
2Université Bordeaux 1, Institut de Mathématiques, UMR 5466 CNRS, 351, cours de la Libération, 33405 Talence cedex, France
3Consiglio Nazionale delle Ricerche, Istituto per le Applicazioni del Calcolo “M. Picone”, Viale del Policlinico 137, I‐00161 Roma, Italy

Tóm tắt

Abstract

We study the asymptotic time behavior of global smooth solutions to general entropy, dissipative, hyperbolic systems of balance laws in m space dimensions, under the Shizuta‐Kawashima condition. We show that these solutions approach a constant equilibrium state in the Lp‐norm at a rate O(t− (m/2)(1 − 1/p)) as t → ∞ for p ∈ [min{m, 2}, ∞]. Moreover, we can show that we can approximate, with a faster order of convergence, the conservative part of the solution in terms of the linearized hyperbolic operator for m ≥ 2, and by a parabolic equation, in the spirit of Chapman‐Enskog expansion in every space dimension. The main tool is given by a detailed analysis of the Green function for the linearized problem. © 2007 Wiley Periodicals, Inc.

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Thông tin xuất bản

Communications on Pure and Applied Mathematics

Tập 60 Số 11

1559-1622

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