Acta mathematica scientia, Series B >
STRONG COMPACTNESS OF APPROXIMATE SOLUTIONS TO DEGENERATE ELLIPTIC-HYPERBOLIC EQUATIONS WITH DISCONTINUOUS FLUX FUNCTION
Received date: 2009-10-25
Online published: 2009-11-20
Supported by
This work was supported by the Research Council of Norway through the projects Nonlinear Problems in Mathematical Analysis; Waves In Fluids and Solids; Outstanding Young Investigators Award (KHK), and the Russian Foundation for Basic Research (grant No.~09-01-00490-a) and DFG project No.~436 RUS 113/895/0-1 (EYuP). This article was written as part of the the international research program
on Nonlinear Partial Differential Equations at the Centre for Advanced Study at the Norwegian Academy of Science and Letters in Oslo during the academic year 2008--09.
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.
Helge Holden , Kenneth H. Karlsen , Darko Mitrovic , Evgueni Yu. Panov . STRONG COMPACTNESS OF APPROXIMATE SOLUTIONS TO DEGENERATE ELLIPTIC-HYPERBOLIC EQUATIONS WITH DISCONTINUOUS FLUX FUNCTION[J]. Acta mathematica scientia, Series B, 2009 , 29(6) : 1573 -1612 . DOI: 10.1016/S0252-9602(10)60004-5
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