Long-time behavior of scalar viscous shock fronts in two dimensions (Q1300112)

The scalar viscous conservation law \[ u_t+f(u)_x+g(u)_y= u_{xx}+ u_{yy},\quad \bigl(f(u)= u^2/2), \] is considered and the long-time behavior, in \(L^1(\mathbb{R}^2)\), of solutions whose initial data are close to a planar shock wave is studied. The authors prove nonlinear stability of the solutions and obtain an expression for the asymptotic form of small perturbations showing also that the long-time behavior is determined by an effective diffusion coefficient depending on forces transverse to the shock front.