Interaction of elementary waves for scalar conservation laws on a bounded domain

Abstract

Abstract This paper is concerned with the interaction of elementary waves on a bounded domain for scalar conservation laws. The structure and large time asymptotic behaviours of weak entropy solution in the sense of Bardos et al . (Comm. Partial Differential Equations 1979; 4 : 1017) are clarified to the initial–boundary problem for scalar conservation laws u t +ƒ( u ) x =0 on (0,1) × (0,∞), with the initial data u ( x ,0)= u 0 (x):= u m and the boundary data u (0, t )= u ‐, u (1, t )= u + , where u ±, u m are constants, which are not equivalent, and ƒ∈C 2 satisfies ƒ′′>0, ƒ(0)=f′(0)=0. We also give some global estimates on derivatives of the weak entropy solution. These estimates play important roles in studying the rate of convergence for various approximation methods to scalar conservation laws. Copyright © 2003 John Wiley & Sons, Ltd.