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October  2011, 29(4): 1393-1404. doi: 10.3934/dcds.2011.29.1393

Asymptotic behaviour of a porous medium equation with fractional diffusion

Luis Caffarelli 1, and  Juan-Luis Vázquez 2,

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

Received  March 2010 Revised  August 2010 Published  December 2010

We consider a porous medium equation with a nonlocal diffusion effect given by an inverse fractional Laplacian operator. The equation is posed in the whole space $\mathbb{R}^n$. In a previous paper we have found mass-preserving, nonnegative weak solutions of the equation satisfying energy estimates. Here we establish the large-time behaviour. We first find selfsimilar nonnegative solutions by solving an elliptic obstacle problem for the pair pressure-density involving the Laplacian, obtaining what we call obstacle Barenblatt solutions. The theory for elliptic fractional problems with obstacles has been recently established. We then use entropy methods to show that the asymptotic behavior of general finite-mass solutions is described after renormalization by these special solutions, which represent a surprising variation of the Barenblatt profiles of the standard porous medium model.
Keywords: Porous medium equation, obstacle problem, fractional Laplacian, asymptotic behavior..
Mathematics Subject Classification: Primary: 35B40, 35K55, 35K65, 35J87, 26A3.
Citation: Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393
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.  Google Scholar

[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.  Google Scholar

[3]

P. Biler, C. Imbert and G. Karch, Fractal porous medium equation,, preprint., ().   Google Scholar

[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.  Google Scholar

[5]

L. A. Caffarelli, The obstacle problem revisited, The Journal of Fourier Analysis and Applications, 4 (1998), 383-402. doi: 10.1007/BF02498216.  Google Scholar

[6]

L. A. Caffarelli, Further regularity for the Signorini problem, Comm. Partial Differential Equations, 4 (1979), 1067-1075. doi: 10.1080/03605307908820119.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

L. A. Caffarelli, F. Soria and J. L. Vázquez, Regularity of solutions of the fractional porous medium flow,, in preparation., ().   Google Scholar

[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.  Google Scholar

[11]

L. A. Caffarelli and J. L. Vázquez, Nonlinear porous medium flow with fractional potential pressure,, \arXiv{1001.0410}., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

A. Friedman, "Variational Principles and Free Boundary Problems," Wiley, New York, 1982.  Google Scholar

[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.  Google Scholar

[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)  Google Scholar

[18]

A. K. Head, Dislocation group dynamics II. Similarity solutions of the continuum approximation, Phil. Mag., 26 (1972), 65-72. doi: 10.1080/14786437208221020.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.  Google Scholar

[22]

J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

P. Biler, C. Imbert and G. Karch, Fractal porous medium equation,, preprint., ().   Google Scholar

[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.  Google Scholar

[5]

L. A. Caffarelli, The obstacle problem revisited, The Journal of Fourier Analysis and Applications, 4 (1998), 383-402. doi: 10.1007/BF02498216.  Google Scholar

[6]

L. A. Caffarelli, Further regularity for the Signorini problem, Comm. Partial Differential Equations, 4 (1979), 1067-1075. doi: 10.1080/03605307908820119.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

L. A. Caffarelli, F. Soria and J. L. Vázquez, Regularity of solutions of the fractional porous medium flow,, in preparation., ().   Google Scholar

[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.  Google Scholar

[11]

L. A. Caffarelli and J. L. Vázquez, Nonlinear porous medium flow with fractional potential pressure,, \arXiv{1001.0410}., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

A. Friedman, "Variational Principles and Free Boundary Problems," Wiley, New York, 1982.  Google Scholar

[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.  Google Scholar

[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)  Google Scholar

[18]

A. K. Head, Dislocation group dynamics II. Similarity solutions of the continuum approximation, Phil. Mag., 26 (1972), 65-72. doi: 10.1080/14786437208221020.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.  Google Scholar

[22]

J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.  Google Scholar

[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.  Google Scholar

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