Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy
Stefano Bianchini, Bernard Hanouzet, Roberto Natalini
Source abstract
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 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.
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