Asymptotic limit of initial boundary value problems for conservation laws with relaxational extensions

We study the boundary layer effect in the small relaxation limit to the equilibrium scalar conservation laws in one space dimension for the relaxation system proposed in [6]. First, it is shown that for initial and boundary data satisfying a strict version of the subcharacteristic condition, there exists a unique global (in time) solution, ( u ε , v ε ), to the relaxation system (1.4) for each ε > 0. The spatial total variation of ( u ε , v ε ) is shown to be bounded independently of ε, and consequently, a subsequence of ( u ε , v ε ) converges to a limit ( u , v ) as ε → 0 + . Furthermore, u ( x , t ) is a weak solution to the scalar conservation law (1.5) and v = f ( u ). Next, we prove that for data that are suitably small perturbations of a nontransonic state, the relaxation limit function satisfies the boundary‐entropy condition (2.11). Finally, the weak solutions to (1.5) with the boundary‐entropy condition (2.11) is shown to be unique. Consequently, the relaxation limit of solutions to (1.4) is unique, and the whole sequence converges to the unique limit. One consequence of our analysis shows that the boundary layer occurs only in the u ‐component in the sense that v ε (0, ·) converges strongly to γ ○ v = f (γ ○ u ), the trace of f ( u ) on the t ‐axis. © 1998 John Wiley & Sons, Inc.