On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients

Discrete and Continuous Dynamical Systems

Volumn 9, Issue 5, 2003, Pages 1081-1104

  Karlsen, Kenneth Hvistendahl ^a   Risebro, Nils Henrik ^b  

^a UNIVERSITY OF BERGEN   (Norway)

^b UNIVERSITY OF OSLO   (Norway)

Author keywords

Degenerate parabolic equation; Entropy solution; Rough coefficient; Stability; Uniqueness

Indexed keywords

EID: 0038784363     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2003.9.1081     Document Type: Article

References (41)

1. H. W. Alt and S. Luckhaus. Quasilinear elliptic-parabolic differential equations. Math. Z., 183(3):311-341, 1983.
2. 1 of degenerate parabolic equations. J. Differential Equations, 119(2):473-502, 1995.
3. 1. I. Étude du problème stationnaire. In Evolution equations (Baton Rouge, LA, 1992), pages 35-62. Dekker, New York, 1995.
4. 1. II. Le problème d'évolution. Ann. Inst. H. Poincaré Anal. Non Linéaire, 12(6):727-761, 1995.
5. F. Bouchut and B. Perthame. Kruzδkov's estimates for scalar conservation laws revisited. Trans. Amer. Math. Soc., 350(7):2847-2870, 1998.
6. φ(u) = 0. J. Math. Pures Appl. (9), 58(2):153-163, 1979.
7. R. Bürger, S. Evje, and K. H. Karlsen. On strongly degenerate convection-diffusion problems modeling sedimentation-consolidation processes. J. Math. Anal. Appl., 247(2):517-556.
8. R. Bürger and W. L. Wendland. Existence, uniqueness, and stability of generalized solutions of an initial-boundary value problem for a degenerating quasilinear parabolic equation. J. Math. Anal. Appl. 218(1):207-239, 1998.
9. M. C. Bustos, F. Concha, R. Bürger, and E. M. Tory. Sedimentation and Thickening: Phe- nomenological Foundation and Mathematical Theory. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999.
10. I. Capuzzo-Dolcetta and B. Perthame. On some analogy between different approaches to first order PDE's with nonsmooth coefficients. Adv. Math. Sci. Appl., 6(2):689-703, 1996.
11. J. Carrillo. On the uniqueness of the solution of the evolution dam problem. Nonlinear Anal., 22(5):573-607, 1994.
12. J. Carrillo. Entropy solutions for nonlinear degenerate problems. Arch. Rational Mech. Anal., 147(4):269-361, 1999.
13. G.-Q. Chen and E. DiBenedetto. Stability of entropy solutions to the Cauchy problem for a class of hyperbolic-parabolic equations. Preprint, 1999. To appear in SIAM J. Math. Anal.
14. B. Cockburn and G. Gripenberg. Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations. J. Differential Equations, 151(2):231-251, 1999.
15. M. S. Espedal and K. H. Karlsen. Numerical solution of reservoir flow models based on large time step operator splitting algorithms. In Filtration in Porous Media and Industrial Applications, Lecture Notes in Mathematics, 1734, Springer.
16. S. Evje, K. H. Karlsen, and N. H. Risebro. A continuous dependence result for nonlinear degenerate parabolic equations with spatially dependent flux function. Proc. Hyp 2000.
17. T. Gimse and N. H. Risebro. Solution of the Cauchy problem for a conservation law with a discontinuous flux function. SIAM J. Math. Anal., 23(3):635-648, 1992.
18. 1 convergence theory. Preprint (in preparation).
19. E. Isaacson and B. Temple. Convergence of the 2 × 2 Godunov method for a general resonant nonlinear balance law. SIAM J. Appl. Math., 55(3):625-640, 1995.
20. K. H. Karlsen and K.-A. Lie. An unconditionally stable splitting scheme for a class of nonlinear parabolic equations. IMA J. Numer. Anal., 19:609-635.
21. K. H. Karlsen and N. H. Risebro. Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients. Math. Model. and Numer. Anal., 35:239-270, 2001.
22. R. A. Klausen and N. H. Risebro. Stability of conservation laws with discontinuous coefficients. J. Differential Equations, 157(1):41-60, 1999.
23. C. Klingenberg and N. H. Risebro. Stability of a resonant system of conservation laws modeling polymer flow with gravitation. J. Differential Equations, 170:344-380, 2001.
24. C. Klingenberg and N. H. Risebro. Convex conservation laws with discontinuous coefficients. Existence, uniqueness and asymptotic behavior. Comm. Partial Differential Equations, 20(11-12):1959-1990, 1995.
25. S. N. Kružkov. First order quasi-linear equations in several independent variables. Math. USSR Sbornik, 10(2):217-243, 1970.
26. N. N. Kuznetsov. Accuracy of some approximative methods for computing the weak solutions of a first-order quasi-linear equation. USSR Comput. Math. and Math. Phys. Dokl., 16(6):105-119, 1976.
27. B. J. Lucier. A moving mesh numerical method for hyperbolic conservation laws. Math. Comp., 46(173):59-69, 1986.
28. O. A. Oleǐnik. Discontinuous solutions of non-linear differential equations. Amer. Math. Soc Transl. Ser. 2, 26:95-172, 1963.
29. 1-contraction and uniqueness for quasilinear elliptic-parabolic equations. J. Differential Equations, 131(1):20-38, 1996.
30. É. Rouvre and G. Gagneux. Solution forte entropique de lois scalaires hyperboliques-paraboliques dégénérées. C. R. Acad. Sci. Paris Sér. I Math., 329(7):599-602, 1999.
31. T. Tassa. Uniqueness of piecewise smooth weak solutions of multidimensional degenerate parabolic equations. J. Math. Anal. Appl., 210(2):598-608, 1997.
32. B. Temple. Global solution of the Cauchy problem for a class of 2 × 2 nonstrictly hyperbolic conservation laws. Adv. in Appl. Math., 3(3):335-375, 1982.
33. J. Towers. Convergence of a difference scheme for conservation laws with a discontinuous flux. SIAM J. Numer. Anal. 38 (2000), no. 2, 681-698.
34. J. A. Trangenstein. Adaptive mesh refinement for wave propagation in nonlinear solids. SIAM J. Sci. Comput., 16(4):819-839, 1995.
35. A. I. Vol'pert. The spaces BV and quasi-linear equations. Math. USSR Sbornik, 2(2):225-267, 1967.
36. A. I. Vol'pert and S. I. Hudjaev. Cauchy's problem for degenerate second order quasilinear parabolic equations. Math. USSR Sbornik, 7(3):365-387, 1969.
37. G. B. Whitham. Linear and nonlinear waves. Wiley-Interscience [John Wiley & Sons], New York, 1974. Pure and Applied Mathematics.
38. x and their applications to the uniqueness of solutions for degenerate quasilinear parabolic equations. Northeastern Math. J., 5(4):395-422, 1989.
39. J. Yin. On the uniqueness and stability of BV solutions for nonlinear diffusion equations. Comm. in Partial Differential Equations, 15(12):1671-1683, 1990.
40. J. Zhao. Uniqueness of solutions for higher dimensional quasilinear degenerate parabolic equation. Chinese Ann. Math, 13B(2):129-136, 1992.
41. J. Zhao and P. Lei. Uniqueness and stability of solutions for Cauchy problem of nonlinear diffusion equations. Sci. China Ser. A, 40(9):917-925, 1997.