Vanishing viscosity solutions of hyperbolic systems of conservation laws with boundary

ABSTACT. We consider the initial-boundary value problem for the quasilinear hyperbolic system (1) u_t + A(t,u) u_x = 0, u(0,x) = u_0(x), u(t,0) = u_b(t), The matrix valued function A(u,t) is strictly hyperbolic and, for any u in an open domain of R^n, the map t --> A(u,t) has finite total variation. The dependence of A w.r.t. time comprehend the case where the speed of the boundary changes. No assumption is made on the relations between the characteristic speeds of A(t,u) and the speed of the boundary. In particular, the boundary may have the same speed of a characteristic family (boundary characteristic) and this family may change in time. Under the assumption that the initial-boundary data u_0, u_b have sufficiently small total variation, and are close to a constant state bar u, we have shown that the solution u^epsilon of the parabolic approximation of (1) (2) u^epsilon_t + A(t,u^epsilon) u^epsilon_x = =epsilon u^epsilon_{xx}, u^epsilon(0,x) = u_0^epsilon (x), u^epsilon(t,0) = u_b^epsilon (t), is globally defined in time and has uniformly bounded total variation. Moreover, it depends Lipschitz continuously on the initial-boundary data, and converges, as epsilon --> 0, to the "vanishing viscosity solution" of (1). An outline of the proofs of these results is contained in this note.