Continuous Dependence for 2×2 Conservation Laws with Boundary

Abstract

on the domain 0.[(t, x) # R : t 0 and x 9(t)], for some boundary profile 9: R [ R. As usual, (1.1) is assumed to be strictly hyperbolic and with each characteristic field either linearly degenerate or genuinely non linear. An initial data u(0, x)=u (x) having sufficiently small total variation is given. We consider two different kinds of boundary conditions along x=9(t) and in both cases we construct a Lipschitzean flow whose trajectories are solutions of an initial-boundary value problem for (1.1). Thus, we prove the continuous dependence of the solution upon the initial data, upon the boundary condition and upon the boundary profile. The existence theory for global BV solutions to the Cauchy Problem for (1.1) goes back to the fundamental paper [15] by Glimm. More recently, in [6], a new approach has been introduced. It relies on the construction of a Lipschitzean semigroup, the Standard Riemann Semigroup (SRS), whose trajectories extend the local standard Lax [17] solutions of Riemann Problems. At present, such a SRS has been constructed in the 2_2 case in [8, 9], while for the general n_n case see [7, 10] and [3] in article no. DE973274