Abstract.
We propose a new notion of weak solutions (dissipative solutions) for non-isotropic, degenerate, second-order, quasi-linear parabolic equations. This class of solutions is an extension of the notion of dissipative solutions for scalar conservation laws introduced by L. C. Evans. We analyze the relationship between the notions of dissipative and entropy weak solutions for non-isotropic, degenerate, second-order, quasi-linear parabolic equations. As an application we prove the strong convergence of a general relaxation-type approximation for such equations.
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Belhadj, M., Gerbeau, J.-F., Perthame, B.: A multiscale colloid transport model with anisotropic degenerate diffusion. Asympt. Anal. 34, 41–54 (2003)
Bouchut, F., Frid, H.: Finite difference schemes with cross-derivatives correctors for multidimensional parabolic systems. Work in preparation.
Carrillo, J.: Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal. 147, 269–361 (1999)
Chen, G.-Q., Perthame, B.: Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations. Ann. Inst. H. Poincaré, Analyse Non-linéaire. 20, 645–668 (2003)
Dafermos, C.M.: Hyperbolic conservation laws in continuum physics. Springer Verlag, GM 325, 1999
Donatelli, D., Marcati, P.: Convergence of singular limits for multi-d similinear hyperbolic systems to parabolic systems. Preprint, 2003
Eymard, R., Gallouët, T., Herbin, R., Michel, A.: Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. Numer. Math. 92, 41–82 (2002)
Karlsen, K.H., Risebro, N.H.: On convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients. M2AN Math. Model. Numer. Anal. 35, 239–270 (2001)
Katsoulakis, M., Tzavaras, A.: Contractive relaxation systems and the scalar multidimensional conservation laws. Comm. PDE 22, 195–233 (1997)
Katsoulakis, M., Tzavaras, A.: Multiscale analysis for interacting particles: relaxation systems and scalar conservation laws. J. Stat. Phys. 96, 715–763 (1999)
Kruzhkov, S.: First order quasilinear equations with several space variables. Mat. Sbornik 123, 228–255 (1970); Engl. Transl. Math. USSR Sb. 10, 217–273 (1970)
Natalini, R.: Recent results on hyperbolic relaxation problems. In: Analysis of systems of conservation laws. Chapman and Hall/CRC, Boca Raton, FL, 1999
Perthame, B.: Kinetic Formulations of Conservation Laws. Oxford Univ. Press, 2002
Portilheiro, M.A.: Weak solutions for equations defined by accretive operators I. Preprint
Portilheiro, M.A.: Weak solutions for equations defined by accretive operators II: relaxation limits. Preprint
Serre, D.: Systèmes hyperboliques de lois de conservation, Parties I et II. Diderot, Paris, 1996
Volpert, A.I., Hudjaev, S.I.: Cauchy’s problem for degenerate second order quasilinear parabolic equations. Mat. Sbornik 78, 374–396 (1969); Engl. Transl. Math. USSR Sb. 7, 365–387 (1969)
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Communicated by L. C. Evans
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Perthame, B., Souganidis, P. Dissipative and Entropy Solutions to Non-Isotropic Degenerate Parabolic Balance Laws. Arch. Rational Mech. Anal. 170, 359–370 (2003). https://doi.org/10.1007/s00205-003-0282-5
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DOI: https://doi.org/10.1007/s00205-003-0282-5