Vanishing Viscosity Solutions of Hyperbolic Systems with Boundary Conditions

with a well defined notion of trace at x = 0. We allow the boundary x = 0 to have the same speed of one of the characteristic speed of the limiting hyperbolic system. This result is achieved by a new decomposition of the solution into travelling profiles and the non characteristic part of boundary profile, the analysis of the interaction of travelling profiles with the boundary profile, the construction of boundary profile when one non linear characteristic field has a speed close to the speed of the boundary, and corresponding solution of the boundary Riemann problem (i.e. (0.2) with initial boundary data u0, ub constant) and the precise analysis of the trace of the solution u to (0.2) at x = 0. In the last part of the paper we show how the analysis can be extended to the case when the total variation of κ is large. A corollary of the above results is the construction of the solutions to (0.1), (0.2) in the case of oscillating boundary, i.e. x ≥ xb(t), where (t, xb(t)) is a smooth curve in R 2 with speed σb(t) = ẋb(t) of bounded total variation.