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Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations. (English) Zbl 1031.35077
Summary: We develop a well-posedness theory for solutions in $$L^1$$ to the Cauchy problem of general degenerate parabolic-hyperbolic equations with non-isotropic nonlinearity. A new notion of entropy and kinetic solutions and a corresponding kinetic formulation are developed which extends the hyperbolic case. The notion of kinetic solutions applies to more general situations than that of entropy solutions; and its advantage is that the kinetic equations in the kinetic formulation are well defined even when the macroscopic fluxes are not locally integrable, so that $$L^1$$ is a natural space on which the kinetic solutions are posed. Based on this notion, we develop a new, simpler, more effective approach to prove the contraction property of kinetic solutions in $$L^1$$, especially including entropy solutions. It includes a new ingredient, a chain rule type condition, which makes it different from the isotropic case.

##### MSC:
 35K65 Degenerate parabolic equations 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 35D05 Existence of generalized solutions of PDE (MSC2000) 35K15 Initial value problems for second-order parabolic equations
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##### References:
 [1] Bénilan, P.; Carrillo, J.; Wittbold, P., Renormalized entropy solutions of scalar conservation laws, Ann. sc. norm. sup. Pisa, 29, 313-327, (2000) · Zbl 0965.35021 [2] Bouchut, F., Renormalized solutions to the Vlasov equation with coefficients of bounded variation, Arch. ration. mech. anal., 157, 75-90, (2001) · Zbl 0979.35032 [3] Brenier, Y., Résolution d’équations d’évolution quasilinéaires en dimensions N d’espace à l’aide d’équations linéaires en dimensions N+1, J. differential equations, 50(3), 375-390, (1982) · Zbl 0549.35055 [4] Brézis, H.; Crandall, M.G., Uniqueness of solutions of the initial-value problem for ut−δϕ(u)=0, J. math. pure appl. (9), 58, 2, 153-163, (1979) · Zbl 0408.35054 [5] Bustos, M.C.; Concha, F.; Bürger, R.; Tory, E.M., Sedimentation and thickening: phenomenological foundation and mathematical theory, (1999), Kluwer Academic Dordrecht · Zbl 0936.76001 [6] Carrillo, J., Entropy solutions for nonlinear degenerate problems, Arch. rational mech. anal., 147, 269-361, (1999) · Zbl 0935.35056 [7] Chavent, G.; Jaffre, J., Mathematical models and finite elements for reservoir simulation, (1986), North Holland Amsterdam · Zbl 0603.76101 [8] Chen, G.-Q.; DiBenedetto, E., Stability of entropy solutions to the Cauchy problem for a class of nonlinear hyperbolic-parabolic equations, SIAM J. math. anal., 33, 751-762, (2001) · Zbl 1027.35080 [9] Cockburn, B.; Dawson, C., Some extensions of the local discontinuous Galerkin method for convection-diffusion equations in multidimension, (), 225-238 · Zbl 0960.65107 [10] DiBenedetto, E., Continuity of weak solutions to certain singular parabolic equations, Ann. mat. pura appl., 130, 131-176, (1982) · Zbl 0503.35018 [11] Douglis, J.; Dupont, T.; Ewing, R., Incomplete iteration for time-stepping a Galerkin method for a quasilinear parabolic problem, SIAM J. numer. anal., 16, 503-522, (1979) · Zbl 0411.65064 [12] Espedal, M.S.; Fasano, A.; Mikelić, A., Filtration in porous media and industrial applications, Lecture notes in math., 1734, (2000), Springer-Verlag Berlin [13] Eymard, R.; Gallouët, T.; Herbin, R., Existence and uniqueness of the entropy solution to a nonlinear hyperbolic equation, Chinese ann. math. ser. B, 16, 1-14, (1995) · Zbl 0830.35077 [14] R. Eymard, T. Gallouët, R. Herbin, A. Michel, Convergence of a finite volume scheme for nonlinear degenerate parabolic equations, Preprint, 2001 [15] Gilbarg, D.; Trudinger, N., Elliptic partial differential equations of second order, (1983), Springer-Verlag Berlin · Zbl 0562.35001 [16] Gilding, B.H., Improved theory for a nonlinear degenerate parabolic equation, Ann. scuola norm. sup. Pisa cl. sci., 16, 4, 165-224, (1989) · Zbl 0702.35140 [17] Karlsen, K.H.; Risebro, N.H., On convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients, M2AN, 35, 2, 239-270, (2001) · Zbl 1032.76048 [18] Kruzhkov, S., First order quasilinear equations with several space variables, Mat. sbornik, Math. USSR sb., 10, 217-273, (1970), Engl. Transl.: · Zbl 0215.16203 [19] Lions, P.-L.; Perthame, B.; Tadmor, E., Formulation cinétique des lois de conservation scalaires multidimensionnelles, C. R. acad. sci. Paris, Série I math., 312, 97-102, (1991) · Zbl 0729.49007 [20] Lions, P.-L.; Perthame, B.; Tadmor, E., A kinetic formulation of multidimensional scalar conservation laws and related equations, J. amer. math. soc., 7, 169-191, (1994) · Zbl 0820.35094 [21] Perthame, B., Uniqueness and error estimates in first order quasilinear conservation laws via the kinetic entropy defect measure, J. math. pure appl., 77, 1055-1064, (1998) · Zbl 0919.35088 [22] B. Perthame, Kinetic Formulations of Conservation Laws, Oxford Univ. Press, Oxford (to appear) · Zbl 1362.35342 [23] Perthame, B.; Bouchut, F., Kruzhkov’s estimates for scalar conservation laws revisited, Trans. amer. math. soc., 350, 2847-2870, (1998) · Zbl 0955.65069 [24] Volpert, A.I.; Hudjaev, S.I., Cauchy’s problem for degenerate second order quasilinear parabolic equations, Mat. sbornik, Math. USSR sb., 7, 3, 365-387, (1969), Engl. Transl.:
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