Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations
Annales de l'I.H.P. Analyse non linéaire, Tome 20 (2003) no. 4, p. 645-668
@article{AIHPC_2003__20_4_645_0,
     author = {Chen, Gui-Qiang and Perthame, Beno\^\i t},
     title = {Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {20},
     number = {4},
     year = {2003},
     pages = {645-668},
     doi = {10.1016/S0294-1449(02)00014-8},
     zbl = {1031.35077},
     mrnumber = {1981403},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2003__20_4_645_0}
}
Chen, Gui-Qiang; Perthame, Benoît. Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations. Annales de l'I.H.P. Analyse non linéaire, Tome 20 (2003) no. 4, pp. 645-668. doi : 10.1016/S0294-1449(02)00014-8. http://www.numdam.org/item/AIHPC_2003__20_4_645_0/

[1] Bénilan P., Carrillo J., Wittbold P., Renormalized entropy solutions of scalar conservation laws, Ann. Sc. Norm. Sup. Pisa 29 (2000) 313-327. | Numdam | MR 1784177 | Zbl 0965.35021

[2] Bouchut F., Renormalized solutions to the Vlasov equation with coefficients of bounded variation, Arch. Ration. Mech. Anal. 157 (2001) 75-90. | MR 1822415 | 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) (1982) 375-390. | MR 723577 | 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) (1979) 153-163. | Zbl 0408.35054

[5] Bustos M.C., Concha F., Bürger R., Tory E.M., Sedimentation and Thickening: Phenomenological Foundation and Mathematical Theory, Kluwer Academic, Dordrecht, 1999. | MR 1747460 | Zbl 0936.76001

[6] Carrillo J., Entropy solutions for nonlinear degenerate problems, Arch. Rational Mech. Anal. 147 (1999) 269-361. | MR 1709116 | Zbl 0935.35056

[7] Chavent G., Jaffre J., Mathematical Models and Finite Elements for Reservoir Simulation, North Holland, Amsterdam, 1986. | 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 (2001) 751-762. | MR 1884720 | Zbl 1027.35080

[9] Cockburn B., Dawson C., Some extensions of the local discontinuous Galerkin method for convection-diffusion equations in multidimension, in: MAFELAP 1999 (Uxbridge), The Mathematics of Finite Elements and Applications, 10, Elsevier, Oxford, 1999, pp. 225-238. | MR 1801979 | Zbl 0960.65107

[10] Dibenedetto E., Continuity of weak solutions to certain singular parabolic equations, Ann. Mat. Pura Appl. 130 (1982) 131-176. | MR 663969 | 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 (1979) 503-522. | MR 530483 | Zbl 0411.65064

[12] Espedal M.S., Fasano A., Mikelić A., Filtration in Porous Media and Industrial Applications, Lecture Notes in Math., 1734, Springer-Verlag, Berlin, 2000. | Zbl 0954.00053

[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 (1995) 1-14. | MR 1338923 | 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. | MR 1917365

[15] Gilbarg D., Trudinger N., Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. | MR 737190 | Zbl 0562.35001

[16] Gilding B.H., Improved theory for a nonlinear degenerate parabolic equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 16 (4) (1989) 165-224. | Numdam | MR 1041895 | 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, 35 (2) (2001) 239-270. | Numdam | MR 1825698 | Zbl 1032.76048

[18] Kruzhkov S., First order quasilinear equations with several space variables, Mat. Sbornik 123 (1970) 228-255, Engl. Transl.: , Math. USSR Sb. 10 (1970) 217-273. | 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 (1991) 97-102. | MR 1086510 | 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 (1994) 169-191. | MR 1201239 | 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 (1998) 1055-1064. | MR 1661021 | Zbl 0919.35088

[22] B. Perthame, Kinetic Formulations of Conservation Laws, Oxford Univ. Press, Oxford (to appear). | MR 2064166 | Zbl 1030.35002

[23] Perthame B., Bouchut F., Kruzhkov's estimates for scalar conservation laws revisited, Trans. Amer. Math. Soc. 350 (1998) 2847-2870. | MR 1475677 | Zbl 0955.65069

[24] Volpert A.I., Hudjaev S.I., Cauchy's problem for degenerate second order quasilinear parabolic equations, Mat. Sbornik 78 (120) (1969) 374-396, Engl. Transl.: , Math. USSR Sb. 7 (3) (1969) 365-387. | MR 264232 | Zbl 0191.11603