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Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy (Altre pubblicazioni)

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Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy (Altre pubblicazioni) (literal)

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S. Bianchini, B. Hanouzet, R. Natalini (2005)
Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy
(literal)

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IAC Report 79 (11/2005); to appear in Communications in Pure and Applied Mathematics (literal)

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We study the asymptotic time behavior of global smooth solutions to general entropy dissipative hyperbolic systems of balance law in $m$ space dimensions, under the Shizuta-Kawashima condition. We show that these solutions approach constant equilibrium state in the $L^p$-norm at a rate $O(t^{-\frac{m}{2}(1-\frac{1}{p})})$, as $t\to\infty$, for $p\in [\min{\{m,2\}},\infty]$. 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 $m\geq 2$, and by a parabolic equation, in the spirit of Chapman-Enskog expansion. The main tool is given by a detailed analysis of the Green function for the linearized problem. (literal)

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