Propagation of viscous shock waves away from the boundary

We study the propagation of shock waves away from the boundary for viscous conservation law. Our main purpose is to obtain pointwise description of the perturbation of the shock profile. We show that there are different convergence rates for the region between the boundary and the shock and the region ahead of the shock. The dependence of these rates on the shock strength, viscosity, and initial perturbation is studied. There are two mechanisms which govern the solution behavior: the compressibility of the shock and the presence of the boundary. We introduce an iteration scheme to decouple these two effects. Thus near the boundary we use the Green's function for the initial-boundary value problem of the equation linearized around the boundary value; away from the boundary we use the Green's function for the initial-value problem of the equation linearized around the shock profile. To focus on our main ideas, we study the Burgers equation, for which the Green's functions have explicit forms. Our new approach should be applicable to more general situations such as the system of viscous conservation laws.