INITIAL-BOUNDARY VALUE-PROBLEM FOR CONSERVATION-LAWS

This paper concerns the initial boundary value problems for some syste ms of quasilinear hyperbole conservation laws in the space of bounded measurable functions. The main assumption is that the system under stu dy admits a convex entropy extension. It is proved that then any twice ly differentiable entropy fluxes have traces on the boundary if the bo unded solutions are generated by either Godunov schemes or by suitable viscous approximations. Furthermore, in the case that the weak interi or solutions are generated by Godunov schemes, any Lipschitz continuou s entropy fluxes corresponding to convex entropies have traces on the boundary and the traces are bounded above by computable numerical boun dary values. This in particular gives a trace formula for the Aux func tions in terms of the numerical boundary data. We also investigate the formulation of boundary conditions for systems of hyperbolic conserva tion laws. It is shown that the set of expected boundary values derive d from the viscous approximation contains the one derived in terms of the boundary Riemann problems, and the converse is not true in general . The general theory is then applied to some specific examples. First, several new facts are obtained for convex scalar conservation laws. F or example, we give example which shaw that Godunov schemes produce nu merical boundary layers. It is shown that any continuous functions of density have traces on the boundary (instead of only entropy fluxes). We also obtain interior and boundary regularity of the weak solutions for bounded measurable initial and boundary data. A generalized Oleini k entropy condition is also obtained. Next, we prove the existence of a weak solution to the initial-boundary value problem for a family of 2 x 2 quadratic system with a uniformly characteristic boundary condit ion.