Asymptotic stability of rarefraction waves for 2\(\times 2\) viscous hyperbolic conservation laws - the two-modes case
The author continues his study on the asymptotic behaviour toward rarefraction waves of the solution for the initial value problem for a \(2\times 2\) system of viscous hyperbolic conservation laws of the form \(u_ t+(f(u))_ x=(B(u)u_ x)_ x,\) \(u\in \Omega \subset R^ 2\), \(x\in R^ 1\), \(t>0\), with initial data \(u(x,0)=u_ 0(x)\), \(x\in R^ 1\), satisfying \(\lim_{x\to \pm \infty}u_ 0(x)=u_{\pm}.\) The corresponding inviscid problem is \(u_ t+(f(u))_ x=0,\) \(x\in R^ 1\), \(t\geq 0\) with Riemann initial data \(u(x,0)=u_ -\) if \(x<0\), and \(=u_+\) if \(x>0\). The main result shown in the paper states that when the initial data is a small perturbation of a weak rarefraction wave (a linear superposition of a 1-rarefraction wave and a 2-rarefraction wave) for the inviscid problem, then the solution of the Cauchy problem for the viscous system globally exists and tends to the rarefraction wave.
scientific article; zbMATH DE number 4156976