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Strong compactness of approximate solutions to degenerate elliptic-hyperbolic equations with discontinuous flux function. (English) Zbl 1212.35166
Summary: 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 one can prove convergence of various types of approximate solutions, such as those generated by the vanishing viscosity method and numerical schemes.

35J70 Degenerate elliptic equations
35K65 Degenerate parabolic equations
35L65 Hyperbolic conservation laws
35B25 Singular perturbations in context of PDEs
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