EDP Sciences logo
Subscriber Authentication Point
Search
Menu
All issues Volume 50 / No 1 (January-February 2016) ESAIM: M2AN, 50 1 (2016) 135-162 References
Free Access
Issue
ESAIM: M2AN
Volume 50, Number 1, January-February 2016
Page(s) 135 - 162
DOI https://doi.org/10.1051/m2an/2015031
Published online 16 November 2015
  1. N.D. Alikakos and R. Rostamian, Large time behavior of solutions of Neumann boundary value problem for the porous medium equation. Indiana Univ. Math. J. 30 (1981) 749–785. [CrossRef] [MathSciNet] [Google Scholar]
  2. T. Arbogast, M. Wheeler and N.-Y. Zhang, A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous medium. SIAM J. Numer. Anal. 33 (1996) 1669–1687. [CrossRef] [MathSciNet] [Google Scholar]
  3. A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations. Commun. Partial Differ. Equ. 26 (2001) 43–100. [Google Scholar]
  4. J. Barrett, J. Blowey and H. Garcke, Finite element approximation of a fourth order nonlinear degenerate parabolic equation. Numer. Math. 80 (1998) 525–556. [CrossRef] [MathSciNet] [Google Scholar]
  5. W. Beckner, A generalized Poincaré inequality for Gaussian measures. Proc. Amer. Math. Soc. 105 (1989) 397–400. [MathSciNet] [Google Scholar]
  6. J. Berryman, Extinction time for fast diffusion. Phys. Lett. A 72 (1979) 107–110. [CrossRef] [Google Scholar]
  7. M. Bessemoulin-Chatard, C. Chainais–Hillairet and F. Filbet, On discrete functional inequalities for some finite volume schemes. IMA J. Numer. Anal. 35 (2015) 1125–1149. [CrossRef] [MathSciNet] [Google Scholar]
  8. M. Bonforte and G. Grillo, Asymptotics of the porous media equations via Sobolev inequalities. J. Funct. Anal. 225 (2005) 33–62. [CrossRef] [MathSciNet] [Google Scholar]
  9. M. Bonforte, J. Dolbeault, G. Grillo and J.L. Vázquez, Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities. Proc. Natl. Acad. Sci. USA 107 (2010) 16459–16464. [CrossRef] [MathSciNet] [Google Scholar]
  10. J.A. Carrillo and G. Toscani, Asymptotic L1-decay of solutions of the porous medium equation to selfsimilarity. Indiana Univ. Math. J. 49 (2000) 113–141. [CrossRef] [MathSciNet] [Google Scholar]
  11. J.A. Carrillo, A. Jüngel, P. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. Monatsh. Math. 133 (2001) 1–82. [CrossRef] [MathSciNet] [Google Scholar]
  12. J.A. Carrillo, A. Jüngel and S. Tang, Positive entropic schemes for a nonlinear fourth-order equation. Discrete Contin. Dyn. Syst. B 3 (2003) 1–20. [Google Scholar]
  13. J.A. Carrillo, J. Dolbeault, I. Gentil, and A. Jüngel, Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations. Discr. Cont. Dyn. Syst. B 6 (2006) 1027–1050. [CrossRef] [Google Scholar]
  14. F. Cavalli, G. Naldi, G. Puppo and M. Semplice, High-order relaxation schemes for nonlinear degenerate diffusion problems. SIAM J. Numer. Anal. 45 (2007) 2098–2119. [CrossRef] [Google Scholar]
  15. M. Del Pino and J. Dolbeault, Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions. J. Math. Pures Appl. 81 (2002) 847–875. [CrossRef] [MathSciNet] [Google Scholar]
  16. J. Denzler and R. McCann, Fast diffusion to self-similarity: complete spectrum, long-time asymptotics, and numerology. Arch. Ration. Mech. Anal. 175 (2005) 301–342. [CrossRef] [Google Scholar]
  17. J. Dolbeault, I. Gentil, A. Guillin and F.-Y. Wang, Lq-functional inequalities and weighted porous media equations. Potential Anal. 28 (2008) 35–59. [CrossRef] [MathSciNet] [Google Scholar]
  18. J. Dolbeault, B. Nazaret and G. Savaré, On the Bakry-Emery criterion for linear diffusions and weighted porous media equations. Commun. Math. Sci. 6 (2008) 477–494. [CrossRef] [Google Scholar]
  19. C. Ebmayer and W.W. Liu, Finite element approximation of the fast diffusion and the porous medium equations. SIAM J. Numer. Anal. 46 (2008) 2393–2410. [CrossRef] [Google Scholar]
  20. F. Edgeworth, On the probable errors of frequency-constants (contd.). J. Roy. Stat. Soc. 71 (1908) 499–512. [CrossRef] [Google Scholar]
  21. R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods. In Vol. 7 of Handbook of Numerical Analysis, edited by P.G. Ciarlet and J.L. Lions. North-Holland, Amsterdam (2000) 713–1020. [Google Scholar]
  22. 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. [CrossRef] [MathSciNet] [Google Scholar]
  23. R. Eymard, M. Gutnic and D. Hilhorst, The finite Volume Method for an elliptic-parabolic equation. Proc. of the Algoritmy’97 Conference on Scientific Computing (Zuberec). Acta Math. Univ. Comenian. (N.S.) 67 (1998) 181–195. [Google Scholar]
  24. A. Glitzky and K. Gärtner, Energy estimates for continuous and discretized electro-reaction-diffusion systems. Nonlin. Anal. 70 (2009) 788–805. [CrossRef] [Google Scholar]
  25. G. Grillo and M. Muratori, Sharp short and long-time asymptotics for solutions to the Neumann problem for the porous media equation. J. Differ. Equ. 254 (2013) 2261–2288. [CrossRef] [Google Scholar]
  26. G. Grillo and M. Muratori, Sharp asymptotics for the porous media equation in low dimensions via Gagliardo–Nirenberg inequalities. Riv. Mat. Univ. Parma 5 (2014) 15–38. [Google Scholar]
  27. L. Gross, Logarithmic Sobolev inequalities. Amer. J. Math. 97 (1975) 1061–1083. [Google Scholar]
  28. G. Grün and M. Rumpf, Nonnegativity preserving convergent schemes for the thin film equation. Numer. Math. 87 (2000) 113–152. [CrossRef] [MathSciNet] [Google Scholar]
  29. A. Jüngel and D. Matthes, An algorithmic construction of entropies in higher-order nonlinear PDEs. Nonlinearity 19 (2006) 633–659. [CrossRef] [Google Scholar]
  30. A. Jüngel and D. Matthes, The Derrida–Lebowitz–Speer–Spohn equation: existence, non-uniqueness, and decay rates of the solutions. SIAM J. Math. Anal. 39 (2008) 1996–2015. [CrossRef] [MathSciNet] [Google Scholar]
  31. A. Jüngel and R. Pinnau, A positivity preserving numerical scheme for a nonlinear fourth-order parabolic equation. SIAM J. Numer. Anal. 39 (2001) 385–406. [CrossRef] [MathSciNet] [Google Scholar]
  32. J. King, Interacting dopant diffusions in crystalline silicon. SIAM J. Appl. Math. 48 (1988) 405–415. [CrossRef] [Google Scholar]
  33. R. Latala and K. Oleszkiewicz, Between Sobolev and Poincaré. Lect. Notes Math. 1745 (2000) 147–168. [CrossRef] [Google Scholar]
  34. M. Ledoux, On Talagrand’s deviation inequalities for product measures. ESAIM: PS 1 (1996) 63–87. [CrossRef] [EDP Sciences] [Google Scholar]
  35. E. Lieb and M. Loss, Analysis, 2nd edition. Amer. Math. Soc. Providence (2010). [Google Scholar]
  36. P.-L. Lions and S. Mas-Gallic, Une méthode particulaire déterministe pour des équations diffusives non linéaires. C. R. Acad. Sci. Paris Sér. I Math 332 (2001) 369–376. [CrossRef] [MathSciNet] [Google Scholar]
  37. A. Michel and J. Vovelle, Entropy formulation for parabolic degenerate equations with general Dirichlet boundary conditions and application to the convergence of FV methods. SIAM J. Numer. Anal. 41 (2003) 2262–2293. [CrossRef] [MathSciNet] [Google Scholar]
  38. M. Ohlberger, A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations. Math. Mod. Numer. Anal. 35 (2001) 355–387. [Google Scholar]
  39. M. Rose, Numerical methods for flows through porous media I. Math. Comput. 40 (1983) 435–467. [CrossRef] [MathSciNet] [Google Scholar]
  40. C. Seis, Long-time asymptotics for the porous medium equation: the spectrum of the linearized operator. J. Differ. Equ. 256 (2014) 1191–1223. [CrossRef] [Google Scholar]
  41. O. Shisha, On the discrete version of Wirtinger’s inequality. Amer. Math. Monthly 80 (1973) 755–760. [CrossRef] [MathSciNet] [Google Scholar]
  42. J.L. Vázquez, Smoothing and Decay Estimates for Nonlinear Parabolic Equations of Porous Medium Type. Vol. 33 of Oxford Lect. Ser. Math. Appl. Oxford University Press (2006). [Google Scholar]
  43. J.L. Vázquez, The Porous Medium Equation. Mathematical Theory. Oxford University Press, Oxford (2007). [Google Scholar]
  44. Z. Wu, J. Yin and C. Wang, Elliptic and Parabolic Equations. World Scientific, Singapore (2006). [Google Scholar]
  45. L. Zhornitskaya and A. Bertozzi, Positivity-preserving numerical schemes for lubrication-type equations. SIAM J. Numer. Anal. 37 (2000) 523–555. [CrossRef] [MathSciNet] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you