Wave-front tracking for the equations of isentropic gas dynamics

Asakura, Fumioki

doi:10.1090/S0033-569X-04-00935-8

Author: Fumioki Asakura
Journal: Quart. Appl. Math. 63 (2005), 20-33
MSC (2000): Primary 35L65, 35L67; Secondary 76N10, 76N15
DOI: https://doi.org/10.1090/S0033-569X-04-00935-8
Published electronically: December 13, 2004
MathSciNet review: 2126567
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Abstract: We study the $2 \times 2$ system of conservation laws of the form $v_t - u_x = u_t + p(v)_x =0, \ p = k^2v^{-\gamma } (\gamma \geq 1)$, which are the model equations of isentropic gas dynamics. Weak global in time solutions are obtained by Nishida-Smoller (CPAM 1973) provided $(\gamma - 1)$ times the total variation of the initial data is sufficiently small. The aim of this paper is to give an alternative proof by using the Dafermos-Bressan-Risebro wave-front tracking scheme. We obtain new estimates of the total amount of interactions, which also imply the asymptotic decay of the solution. The main idea is to define appropriate amplitude to the path that is a continuation of shock fronts.

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