L 1-Framework for continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations

Transactions of the American Mathematical Society

Volumn 358, Issue 3, 2006, Pages 937-963

  Chen, Gui Qiang ^a   Karlsen, Kenneth H ^b  

^a NORTHWESTERN UNIVERSITY   (United States)

^b UNIVERSITY OF OSLO   (Norway)

Author keywords

Continuous dependence; Degenerate parabolic equations; Difference schemes; Entropy solutions; Error estimates; Kinetic formulation; L 1 framework; Quasilinear, anisotropic; Vanishing viscosity

Indexed keywords

EID: 33644765655     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9947-04-03689-X     Document Type: Review

References (32)

1. D. Aregba-Driollet, R. Natalini, and S. Tang, Explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems, Math. Comp. 73 (2004), 63-94. MR2034111
2. 1. II. Le problème d'évolution, Ann. Inst. H. Poincaré Anal. Non Linéaire, 12 (1995), 727-761. MR1360542 (96m:35135)
3. F. Bouchut and B. Perthame, Kružkov's estimates for scalar conservation laws revisited, Trans. Amer. Math. Soc. 350 (1998), 2847-2870. MR1475677 (98m:65156)
4. F. Bouchut, F. R. Guarguaglini, and R. Natalini, Diffusive BGK approximations for non-linear multidimensional parabolic equations, Indiana Univ. Math. J. 49 (2000), 723-749. MR1793689 (2001k:35162)
5. R. Bürger, S. Evje, and K. H. Karlsen, On strongly degenerate convection-diffusion problems modeling sedimentation-consolidation processes, J. Math. Anal. Appl. 247 (2000), 517-556. MR1769093 (2001d:35110)
6. M. C. Bustos, F. Concha, R. Bürger, and E. M. Tory, Sedimentation and Thickening: Phenomenological Foundation and Mathematical Theory, Kluwer Academic Publishers: Dordrecht, The Netherlands, 1999. MR1747460 (2002b:76101)
7. J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Rational Mech. Anal. 147 (1999), 269-361. MR1709116 (2000m:35132)
8. G.-Q. Chen and E. DiBenedetto, Stability of entropy solutions to the Cauchy problem for a class of hyperbolic-parabolic equations, SIAM J. Math. Anal. 33 (2001), 751-762. MR1884720 (2002m:35125)
9. G.-Q. Chen and B. Perthame, Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 645-668. MR1981403 (2004c:35235)
10. B. Cockburn and P.-A. Gremaud, A priori estimates for numerical methods for scalar conservation laws, Part I: The general approach, Math. Comp. 65 (1996), 533-573. MR1333308 (96g:65089)
11. B. Cockburn and G. Gripenbergm, Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations, J. Diff. Eqs. 151 (1999), 231-251. MR1669570 (2000a:35101)
12. 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 (Cetraro, Italy, 1998), Lecture Notes in Mathematics, 1734 (2000), pp. 9-77, Springer-Verlag: Berlin. MR1816143 (2002a:76115)
13. S. Evjc arid K. H. Karlsen, Monotone difference approximations of BV solutions to degenerate convection-diffusion equations, SIAM J. Numer. Anal. 37 (2000), 1838-1860. MR1766850 (2001g:65110)
14. S. Evje, K. H. Karlsen, and N. H. Risebro, A continuous dependence result for nonlinear degenerate parabolic equations with spatially dependent flux function, In: Int. Series of Numerical Mathematics 140 (2001), 337-346, Birkhäuser-Verlag. MR1882934
15. R. Eymard, T. Gallouët, R. Herbin, and A. Michel, Convergence of a finite volume scheme for nonlinear degenerate parabolic equations, Numer. Math. 92 (2002), 41-82. MR1917365 (2003e:65159)
16. K. H. Karlsen and M. Ohlberger, A note on the uniqueness of entropy solutions of nonlinear degenerate parabolic equations, J. Math. Anal. Appl. 275 (2002), 439-458. MR1941794 (2004a:35137)
17. K. H. Karlsen and N. H. Risebro, Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients, M2AN Math. Model, Numer. Anal. 35 (2001), 239-269. MR1825698 (2002b:35138)
18. K. H. Karlsen and N. H. Risebro, On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients, Discrete and Continuous Dynamical Systems 9 (2003), 1081-1104. MR1974417
19. S. N. Kružkov, First order quasi-linear equations in several independent variables, Math. USSR Sbornik 10 (1970), 217-243.
20. N. N. Kuznctsov, 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 (1976), 105-119.
21. P.-L. Lions, B. Perthame, and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc. 7 (1994), 169-191. MR1201239 (94d:35100)
22. B. J. Lucier, A moving mesh numerical method for hyperbolic conservation laws, Math. Comp. 46 (1986), 59-69. MR0815831 (87m:65141)
23. C. Makridakis and B. Perthame, Optimal rate of convergence for anisotropic vanishing viscosity limit of a scalar balance law, SIAM J. Math. Anal. 34 (2003), 1300-1307. MR2000971
24. C. Mascia, A. Porretta, and A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equations, Arch. Rational Mech. Anal. 163 (2002), 87-124. MR1911095 (2003e:35217)
25. A. Michel and J. Vovelle, A finite volume method for parabolic degenerate problems with general Dirichlet boundary conditions, Preprint, 2002.
26. M. Ohlberger, A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations, M2AN Math. Model. Numer. Anal. 36 (2001), 355-387. MR1825703 (2002a:65142)
27. B. Perthame, Uniqueness and error estimates in first order quasilinear conservation laws via the kinetic entropy defect measure, J. Math. Pures Appl. 77 (1998), 1055-1064. MR1661021 (2000e:35141)
28. É. Rouvre and G. Gagneux, Solution forte entropique de lois scalaires hyperboliques-paraboliques dégénérées, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), 599-602. MR1717116 (2000g:35160)
29. T. Tassa. Uniqueness of piecewise smooth weak solutions of multidimensional degenerate parabolic equations, J. Math. Anal. Appl. 210 (1987), 598-608. MR1453194 (98c:35080)
30. A. I. Vol'pert, The spaces BV and quasi-linear equations. Math. USSR Sbornik 2 (1967), 225-267. MR0216338 (35:7172)
31. A. I. Vol'pert and S. I. Hudjaev, Cauchy's problem for degenerate second order quasilinear parabolic equations, Math. USSR Sbornik 7 (1969), 365-387. MR0264232 (41:8828)
32. x and their applications to the uniqueness of solutions for degenerate quasilinear parabolic equations, Chinese Northeastern Math. J. 5 (1989), 395-422. MR1053519 (91j:35153)