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A biharmonic-modified forward time stepping method for fourth order nonlinear diffusion equations
Asymptotic behaviour of a porous medium equation with fractional diffusion
1. | Department of Mathematics, University of Texas at Austin, 1 University Station, C1200, Austin, TX 78712-1082 |
2. | Departamento de Matemáticas, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid |
References:
[1] |
I. Athanasopoulos, L. A. Caffarelli and S. Salsa, The structure of the free boundary for lower dimensional obstacle problems, Amer. J. Math., 130 (2008), 485-498.
doi: 10.1353/ajm.2008.0016. |
[2] |
G. I. Barenblatt, On self-similar motions of a compressible fluid in a porous medium (Russian), Akad. Nauk SSSR. Prikl. Mat. Meh., 16 (1952), 679-698. |
[3] |
P. Biler, C. Imbert and G. Karch, Fractal porous medium equation,, preprint., ().
|
[4] |
P. Biler, G. Karch and R. Monneau, Nonlinear diffusion of dislocation density and self-similar solutions, Comm. Math. Phys., 294 (2010), 145-168.
doi: 10.1007/s00220-009-0855-8. |
[5] |
L. A. Caffarelli, The obstacle problem revisited, The Journal of Fourier Analysis and Applications, 4 (1998), 383-402.
doi: 10.1007/BF02498216. |
[6] |
L. A. Caffarelli, Further regularity for the Signorini problem, Comm. Partial Differential Equations, 4 (1979), 1067-1075.
doi: 10.1080/03605307908820119. |
[7] |
L. A. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.
doi: 10.1007/s00222-007-0086-6. |
[8] |
L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Diff. Eqns., 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[9] |
L. A. Caffarelli, F. Soria and J. L. Vázquez, Regularity of solutions of the fractional porous medium flow,, in preparation., ().
|
[10] |
L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2), 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[11] |
L. A. Caffarelli and J. L. Vázquez, Nonlinear porous medium flow with fractional potential pressure,, \arXiv{1001.0410}., ().
|
[12] |
J. A. Carrillo and G. Toscani, Asymptotic $L^1$-decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J., 49 (2000), 113-142.
doi: 10.1512/iumj.2000.49.1756. |
[13] |
J. A. Carrillo, A. Jngel, P. A. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., 133 (2001), 1-82.
doi: 10.1007/s006050170032. |
[14] |
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.
doi: 10.1016/S0021-7824(02)01266-7. |
[15] |
A. Friedman, "Variational Principles and Free Boundary Problems," Wiley, New York, 1982. |
[16] |
A. Friedman and S. Kamin, The asymptotic behavior of gas in an $N$-dimensional porous medium, Trans. Amer. Math. Soc., 262 (1980), 551-563. |
[17] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Classics in Mathematics, Springer, Berlin, 2001. (reprint of the 1998 edition) |
[18] |
A. K. Head, Dislocation group dynamics II. Similarity solutions of the continuum approximation, Phil. Mag., 26 (1972), 65-72.
doi: 10.1080/14786437208221020. |
[19] |
N. S. Landkof, "Foundations Of Modern Potential Theory," Die Grundlehren der mathematischen Wissenschaften, Band 180 (translated from the Russian by A. P. Doohovskoy), Springer, New York, 1972. |
[20] |
L. E. Silvestre, Hölder estimates for solutions of integro differential equations like the fractional Laplace, Indiana Univ. Math. J., 55 (2006), 1155-1174.
doi: 10.1512/iumj.2006.55.2706. |
[21] |
E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. |
[22] |
J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. |
[23] |
J. L. Vázquez, "Smoothing And Decay Estimates For Nonlinear Diffusion Equations. Equations Of Porous Medium Type," Oxford Lecture Series in Mathematics and its Applications, 33, Oxford University Press, Oxford, 2006. |
show all references
References:
[1] |
I. Athanasopoulos, L. A. Caffarelli and S. Salsa, The structure of the free boundary for lower dimensional obstacle problems, Amer. J. Math., 130 (2008), 485-498.
doi: 10.1353/ajm.2008.0016. |
[2] |
G. I. Barenblatt, On self-similar motions of a compressible fluid in a porous medium (Russian), Akad. Nauk SSSR. Prikl. Mat. Meh., 16 (1952), 679-698. |
[3] |
P. Biler, C. Imbert and G. Karch, Fractal porous medium equation,, preprint., ().
|
[4] |
P. Biler, G. Karch and R. Monneau, Nonlinear diffusion of dislocation density and self-similar solutions, Comm. Math. Phys., 294 (2010), 145-168.
doi: 10.1007/s00220-009-0855-8. |
[5] |
L. A. Caffarelli, The obstacle problem revisited, The Journal of Fourier Analysis and Applications, 4 (1998), 383-402.
doi: 10.1007/BF02498216. |
[6] |
L. A. Caffarelli, Further regularity for the Signorini problem, Comm. Partial Differential Equations, 4 (1979), 1067-1075.
doi: 10.1080/03605307908820119. |
[7] |
L. A. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.
doi: 10.1007/s00222-007-0086-6. |
[8] |
L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Diff. Eqns., 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[9] |
L. A. Caffarelli, F. Soria and J. L. Vázquez, Regularity of solutions of the fractional porous medium flow,, in preparation., ().
|
[10] |
L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2), 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[11] |
L. A. Caffarelli and J. L. Vázquez, Nonlinear porous medium flow with fractional potential pressure,, \arXiv{1001.0410}., ().
|
[12] |
J. A. Carrillo and G. Toscani, Asymptotic $L^1$-decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J., 49 (2000), 113-142.
doi: 10.1512/iumj.2000.49.1756. |
[13] |
J. A. Carrillo, A. Jngel, P. A. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., 133 (2001), 1-82.
doi: 10.1007/s006050170032. |
[14] |
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.
doi: 10.1016/S0021-7824(02)01266-7. |
[15] |
A. Friedman, "Variational Principles and Free Boundary Problems," Wiley, New York, 1982. |
[16] |
A. Friedman and S. Kamin, The asymptotic behavior of gas in an $N$-dimensional porous medium, Trans. Amer. Math. Soc., 262 (1980), 551-563. |
[17] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Classics in Mathematics, Springer, Berlin, 2001. (reprint of the 1998 edition) |
[18] |
A. K. Head, Dislocation group dynamics II. Similarity solutions of the continuum approximation, Phil. Mag., 26 (1972), 65-72.
doi: 10.1080/14786437208221020. |
[19] |
N. S. Landkof, "Foundations Of Modern Potential Theory," Die Grundlehren der mathematischen Wissenschaften, Band 180 (translated from the Russian by A. P. Doohovskoy), Springer, New York, 1972. |
[20] |
L. E. Silvestre, Hölder estimates for solutions of integro differential equations like the fractional Laplace, Indiana Univ. Math. J., 55 (2006), 1155-1174.
doi: 10.1512/iumj.2006.55.2706. |
[21] |
E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. |
[22] |
J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. |
[23] |
J. L. Vázquez, "Smoothing And Decay Estimates For Nonlinear Diffusion Equations. Equations Of Porous Medium Type," Oxford Lecture Series in Mathematics and its Applications, 33, Oxford University Press, Oxford, 2006. |
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