Asymptotic stability of rarefaction waves for 2\(\times 2\) viscous hyperbolic conservation laws
We study the asymptotic behavior toward rarefaction waves of the solutions of a general \(2\times 2\) hyperbolic conservation laws with positive viscosity matrix of the form \(u_ t+f(u)_ x=(B(u)u_ x)_ x,\) \(x\in {\mathbb{R}}^ 1\), \(t>0\), \(u\in {\mathbb{R}}^ 2\), with initial data \(u(x,0)=u_ 0(x)\), \(x\in {\mathbb{R}}^ 1\), satisfying \(\lim_{x\to \pm \infty}u_ 0(x)=u_{\pm}\). Under the general assumptions that the system is strictly hyperbolic, genuinely nonlinear, and strongly coupled, we prove that if the initial data is close to a constant state and its values at \(\pm \infty\) lie on the kth rarefaction wave curve for the corresponding hyperbolic conservation laws, then the solution of the viscous problem tends to the rarefaction wave determined by these states. In a following paper by the author [J. Differ. Equations 78, No.2, 191- 219 (1989)], we show that when the initial data is a small perturbation of a weak rarefaction wave (a linear superposition of a 1-rarefaction wave and a 2-rarefaction wave) for the corresponding inviscid hyperbolic conservation laws, then the solution of the Cauchy problem for the viscous system exists and tends to the rarefaction wave. In summary, above two papers show that any weak rarefaction wave for \(2\times 2\) system is nonlinear stable. In a recent paper by \textit{T. Liu} and the author [Commun. Math. Phys. 118, No.3, 451-465 (1988)], above results have been extended to the compressible Navier-Stokes equations, it is shown that Navier-Stokes equations and Euler equations are time- asymptotically equivalent on the level of expansion waves. The nonlinear stability of planar rarefaction wave for scalar equation in multi-space dimension has also been obtained by the author (to appear in Transactions of AMS).
scientific article; zbMATH DE number 4096035
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