PORTUGALIAEMATHEMATICA
Vol. 58, No. 2, pp. 171-193 (2001)
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PORTUGALIAE MATHEMATICA Vol. 58, No. 2, pp. 171-193 (2001) |
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Measure-Valued Solutions and Well-Posedness of Multi-Dimensional Conservation Laws in A Bounded DomainCezar I. Kondo and Philippe G. LeFlochUniversidade Federal de S\ ao Carlos, Departamento de Matemática,Caixa Postal 676, 13565-905, S\ ao Carlos-SP -- BRAZIL E-mail: dcik@dm.ufscar.br and Centre de Mathématiques Appliquées &\ Centre National de la Recherche Scientifique, U.M.R. 7641, Ecole Polytechnique, 91128 Palaiseau -- FRANCE E-mail: dcik@cmap.polytechnique.fr Philippe G. LeFloch, Centre de Mathématiques Appliquées &\ Centre National de la Recherche Scientifique, U.M.R. 7641, Ecole Polytechnique, 91128 Palaiseau -- FRANCE E-mail: lefloch@cmap.polytechnique.fr Abstract: We propose a general framework to establish the strong convergence of approximate solutions to multi-dimensional conservation laws in a bounded domain, provided uniform bounds on their $L^p$ norm and their entropy dissipation measures are available. To this end, existence, uniqueness, and compactness results are proven in a class of entropy measure-valued solutions, following DiPerna and Szepessy. The new features lie in the treatment of the boundary condition, which we are able to formulate by relying only on an $L^p$ uniform bound. This framework is applied here to prove the strong convergence of diffusive approximations of hyperbolic conservation laws. Keywords: Conservation law; Entropy inequality; Measure-valued solution; Boundary condition; Well-posedness. Classification (MSC2000): 35L65.; 76N10. Full text of the article:
Electronic version published on: 9 Feb 2006. This page was last modified: 27 Nov 2007.
© 2001 Sociedade Portuguesa de Matemática
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