On the stability of viscous shock fronts for certain conservation laws in two-dimensional space
The author investigates the Cauchy problem \[ u_t+f(u)_x=\mu \Delta u,\;u(0,x,y)=u_0(x,y),\^^M(t,x,y)\in \mathbb{R}^+\times \mathbb{R}^2, \] where the initial datum is asymptotically constant: \(u(x,y)\to u_\pm \) as \(x\to \pm \infty \) for any fixed \(y\). Local asymptotic stability of a (possibly shifted) travelling wave solution of the form \(u(t,x,y)=\phi (x-st)\) is proved provided that \(s,\;u_\pm \) satisfy the Rankine-Hugoniot condition and a generalized shock condition. Convexity of the flux is not required.
scientific article; zbMATH DE number 1450177