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

By Stefano Bianchini, B. Hanouzet, Roberto Natalini. 2007

188 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3d4c04792729b74b004feb8eacf1f01f05f4ceba

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Citations in the corpus

Decay estimates for hyperbolic balance laws (2009, methodology): “Moreover, our method is different from that used in [1], where a somewhat long analysis of the Green function for the linearized system is used to derive decay properties of the corresponding linearized problems.”
Large Time Asymptotics for Partially Dissipative Hyperbolic Systems (2011, background): “For instance, in [2], the authors proved more precise decay rates for the high frequency component w1 under (SK), and extended the analysis for nonlinear systems.”