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Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy
Author(s) -
Bianchini Stefano,
Hanouzet Bernard,
Natalini Roberto
Publication year - 2007
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20195
Subject(s) - mathematics , dissipative system , mathematical analysis , norm (philosophy) , entropy (arrow of time) , convex function , regular polygon , hyperbolic partial differential equation , partial differential equation , law , thermodynamics , geometry , physics , political science
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 L p ‐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.