Asymptotic behaviour of a porous medium equation with fractional diffusion

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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

Abstract
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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 - A, 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. 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.
[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. 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. doi: 10.1007/BF02498216.
[6]	L. A. Caffarelli, Further regularity for the Signorini problem,, Comm. Partial Differential Equations, 4 (1979), 1067. 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. 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. 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. 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. 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. 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. doi: 10.1016/S0021-7824(02)01266-7.
[15]	A. Friedman, "Variational Principles and Free Boundary Problems,", Wiley, (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.
[17]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Classics in Mathematics, (2001).
[18]	A. K. Head, Dislocation group dynamics II. Similarity solutions of the continuum approximation,, Phil. Mag., 26 (1972), 65. doi: 10.1080/14786437208221020.
[19]	N. S. Landkof, "Foundations Of Modern Potential Theory,", Die Grundlehren der mathematischen Wissenschaften, (1972).
[20]	L. E. Silvestre, Hölder estimates for solutions of integro differential equations like the fractional Laplace,, Indiana Univ. Math. J., 55 (2006), 1155. doi: 10.1512/iumj.2006.55.2706.
[21]	E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton Mathematical Series, (1970).
[22]	J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,", Oxford Mathematical Monographs, (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 (2006).

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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. 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.
[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. 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. doi: 10.1007/BF02498216.
[6]	L. A. Caffarelli, Further regularity for the Signorini problem,, Comm. Partial Differential Equations, 4 (1979), 1067. 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. 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. 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. 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. 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. 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. doi: 10.1016/S0021-7824(02)01266-7.
[15]	A. Friedman, "Variational Principles and Free Boundary Problems,", Wiley, (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.
[17]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Classics in Mathematics, (2001).
[18]	A. K. Head, Dislocation group dynamics II. Similarity solutions of the continuum approximation,, Phil. Mag., 26 (1972), 65. doi: 10.1080/14786437208221020.
[19]	N. S. Landkof, "Foundations Of Modern Potential Theory,", Die Grundlehren der mathematischen Wissenschaften, (1972).
[20]	L. E. Silvestre, Hölder estimates for solutions of integro differential equations like the fractional Laplace,, Indiana Univ. Math. J., 55 (2006), 1155. doi: 10.1512/iumj.2006.55.2706.
[21]	E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton Mathematical Series, (1970).
[22]	J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,", Oxford Mathematical Monographs, (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 (2006).

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