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VISCOUS SHOCK WAVE TRACING, LOCAL CONSERVATION LAWS, AND POINTWISE ESTIMATES

We introduce a new approach to decompose a system of viscous conservation laws with respect to each characteristic wave structures. Under this new decomposition, the global wave interactions of the system are reduced to coupling of nonlinear waves around constant states outside shock region and a scalar conservation law in the shock region to determine the behavior of local internal shock layers. The behavior is characterized by the motion of the viscous shock fronts. It is analyzed by the local conservation laws. We also introduce generalized diffusion waves to localize waves in initial data. We prove stability of a viscous shock layer of 2×2 system; and obtain the optimal rate of convergence.