Strong compactness of approximate solutions to degenerate elliptic-hyperbolic equations with discontinuous flux function
Helge Holden, Kenneth H. Karlsen, Darko Mitrovic and Evgueni Yu. Panov
Abstract: Under a non-degeneracy condition on the nonlinearities we show that sequences of approximate entropy solutions of mixed elliptic-hyperbolic equations are strongly precompact in the general case of a Caratheodory flux vector. The proofs are based on deriving localization principles for H-measures associated to sequences of measure-valued functions. This main result implies existence of solutions to degenerate parabolic convection-diffusion equations with discontinuous flux. Moreover, it provides a framework in which to prove convergence of various types of approximate solutions, such as those generated by the vanishing viscosity method and numerical schemes.