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September  2011, 4(3): 701-716. doi: 10.3934/krm.2011.4.701

Fast diffusion equations: Matching large time asymptotics by relative entropy methods

Jean Dolbeault 1, and  Giuseppe Toscani 2,

1. 

Ceremade (UMR CNRS no. 7534), Université Paris Dauphine, Place de Lattre de Tassigny, 75775 Paris Cédex 16
2. 

Department of Mathematics at the University of Pavia, via Ferrata 1, 27100 Pavia

Received  May 2010 Revised  June 2011 Published  August 2011

A non self-similar change of coordinates provides improved matching asymptotics of the solutions of the fast diffusion equation for large times, compared to already known results, in the range for which Barenblatt solutions have a finite second moment. The method is based on relative entropy estimates and a time-dependent change of variables which is determined by second moments, and not by the scaling corresponding to the self-similar Barenblatt solutions, as it is usually done.
Keywords: asymptotic expansion, sharp rates, porous media equation, intermediate asymptotics, Barenblatt solutions, Fast diffusion equation, second moment, large time behaviour, optimal constants., Hardy-Poincaré inequalities.
Mathematics Subject Classification: Primary: 35B40; Secondary: 35K55, 39B6.
Citation: Jean Dolbeault, Giuseppe Toscani. Fast diffusion equations: Matching large time asymptotics by relative entropy methods. Kinetic & Related Models, 2011, 4 (3) : 701-716. doi: 10.3934/krm.2011.4.701
References:
[1]

A. Arnold, J. A. Carrillo, L. Desvillettes, J. Dolbeault, A. Jüngel, C. Lederman, P. A. Markowich, G. Toscani and C. Villani, Entropies and equilibria of many-particle systems: an essay on recent research, Monatsh. Math., 142 (2004), 35-43. doi: 10.1007/s00605-004-0239-2.  Google Scholar

[2]

G. I. Barenblatt, On some unsteady motions of a liquid and gas in a porous medium, Akad. Nauk SSSR. Prikl. Mat. Meh., 16 (1952), 67-78.  Google Scholar

[3]

Jean-Philippe Bartier, Adrien Blanchet, Jean Dolbeault and Miguel Escobedo, Improved intermediate asymptotics for the heat equation, Appl. Math. Lett., 24 (2011), 76-81. doi: 10.1016/j.aml.2010.08.020.  Google Scholar

[4]

Adrien Blanchet, Matteo Bonforte, Jean Dolbeault, Gabriele Grillo and Juan-Luis Vázquez, Hardy-Poincaré inequalities and applications to nonlinear diffusions, C. R. Math. Acad. Sci. Paris, 344 (2007), 431-436.  Google Scholar

[5]

Adrien Blanchet, Matteo Bonforte, Jean Dolbeault, Gabriele Grillo and Juan Luis Vázquez, Asymptotics of the fast diffusion equation via entropy estimates, Arch. Ration. Mech. Anal., 191 (2009), 347-385. doi: 10.1007/s00205-008-0155-z.  Google Scholar

[6]

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. doi: 10.1073/pnas.1003972107.  Google Scholar

[7]

Matteo Bonforte, Gabriele Grillo and Juan Luis Vázquez, Special fast diffusion with slow asymptotics: entropy method and flow on a Riemann manifold, Arch. Ration. Mech. Anal., 196 (2010), 631-680. doi: 10.1007/s00205-009-0252-7.  Google Scholar

[8]

Matteo Bonforte and Juan Luis Vazquez, Global positivity estimates and Harnack inequalities for the fast diffusion equation, J. Funct. Anal., 240 (2006), 399-428.  Google Scholar

[9]

M. J. Cáceres and Giuseppe Toscani, Kinetic approach to long time behavior of linearized fast diffusion equations, J. Stat. Phys., 128 (2007), 883-925. doi: 10.1007/s10955-007-9329-6.  Google Scholar

[10]

J. A. Carrillo, M. Di Francesco and G. Toscani, Strict contractivity of the 2-Wasserstein distance for the porous medium equation by mass-centering, Proc. Amer. Math. Soc., 135 (2007), 353-363. doi: 10.1090/S0002-9939-06-08594-7.  Google Scholar

[11]

J. A. Carrillo, A. Jüngel, 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

[12]

J. A. Carrillo, C. Lederman, P. A. Markowich and G. Toscani, Poincaré inequalities for linearizations of very fast diffusion equations, Nonlinearity, 15 (2002), 565-580. doi: 10.1088/0951-7715/15/3/303.  Google Scholar

[13]

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.  Google Scholar

[14]

D. Cordero-Erausquin, B. Nazaret and C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, Adv. Math., 182 (2004), 307-332. doi: 10.1016/S0001-8708(03)00080-X.  Google Scholar

[15]

Panagiota Daskalopoulos and Natasa Sesum, On the extinction profile of solutions to fast diffusion, J. Reine Angew. Math., 622 (2008), 95-119. doi: 10.1515/CRELLE.2008.066.  Google Scholar

[16]

Manuel Del Pino and Jean Dolbeault, Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl. (9), 81 (2002), 847-875.  Google Scholar

[17]

Jochen Denzler and Robert J. McCann, Phase transitions and symmetry breaking in singular diffusion, Proc. Natl. Acad. Sci. USA, 100 (2003), 6922-6925. doi: 10.1073/pnas.1231896100.  Google Scholar

[18]

_____, Fast diffusion to self-similarity: complete spectrum, long-time asymptotics, and numerology, Arch. Ration. Mech. Anal., 175 (2005), 301-342. doi: 10.1007/s00205-004-0336-3.  Google Scholar

[19]

Avner Friedman and Shoshana Kamin, The asymptotic behavior of gas in an $n$-dimensional porous medium, Trans. Amer. Math. Soc., 262 (1980), 551-563.  Google Scholar

[20]

Claudia Lederman and Peter A. Markowich, On fast-diffusion equations with infinite equilibrium entropy and finite equilibrium mass, Comm. Partial Differential Equations, 28 (2003), 301-332.  Google Scholar

[21]

Robert J. McCann and Dejan Slepčev, Second-order asymptotics for the fast-diffusion equation, Int. Math. Res. Not., (2006), 22 pp.  Google Scholar

[22]

William I. Newman, A Lyapunov functional for the evolution of solutions to the porous medium equation to self-similarity. I, J. Math. Phys., 25 (1984), 3120-3123. doi: 10.1063/1.526028.  Google Scholar

[23]

Felix Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.  Google Scholar

[24]

James Ralston, A Lyapunov functional for the evolution of solutions to the porous medium equation to self-similarity. II, J. Math. Phys., 25 (1984), 3124-3127. doi: 10.1063/1.526029.  Google Scholar

[25]

Giuseppe Toscani, A central limit theorem for solutions of the porous medium equation, J. Evol. Equ., 5 (2005), 185-203. doi: 10.1007/s00028-005-0183-1.  Google Scholar

[26]

Juan-Luis Vázquez, Asymptotic behaviour for the porous medium equation posed in the whole space, J. Evol. Equ., 3 (2003), 67-118. doi: 10.1007/s000280300004.  Google Scholar

show all references

References:
[1]

A. Arnold, J. A. Carrillo, L. Desvillettes, J. Dolbeault, A. Jüngel, C. Lederman, P. A. Markowich, G. Toscani and C. Villani, Entropies and equilibria of many-particle systems: an essay on recent research, Monatsh. Math., 142 (2004), 35-43. doi: 10.1007/s00605-004-0239-2.  Google Scholar

[2]

G. I. Barenblatt, On some unsteady motions of a liquid and gas in a porous medium, Akad. Nauk SSSR. Prikl. Mat. Meh., 16 (1952), 67-78.  Google Scholar

[3]

Jean-Philippe Bartier, Adrien Blanchet, Jean Dolbeault and Miguel Escobedo, Improved intermediate asymptotics for the heat equation, Appl. Math. Lett., 24 (2011), 76-81. doi: 10.1016/j.aml.2010.08.020.  Google Scholar

[4]

Adrien Blanchet, Matteo Bonforte, Jean Dolbeault, Gabriele Grillo and Juan-Luis Vázquez, Hardy-Poincaré inequalities and applications to nonlinear diffusions, C. R. Math. Acad. Sci. Paris, 344 (2007), 431-436.  Google Scholar

[5]

Adrien Blanchet, Matteo Bonforte, Jean Dolbeault, Gabriele Grillo and Juan Luis Vázquez, Asymptotics of the fast diffusion equation via entropy estimates, Arch. Ration. Mech. Anal., 191 (2009), 347-385. doi: 10.1007/s00205-008-0155-z.  Google Scholar

[6]

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. doi: 10.1073/pnas.1003972107.  Google Scholar

[7]

Matteo Bonforte, Gabriele Grillo and Juan Luis Vázquez, Special fast diffusion with slow asymptotics: entropy method and flow on a Riemann manifold, Arch. Ration. Mech. Anal., 196 (2010), 631-680. doi: 10.1007/s00205-009-0252-7.  Google Scholar

[8]

Matteo Bonforte and Juan Luis Vazquez, Global positivity estimates and Harnack inequalities for the fast diffusion equation, J. Funct. Anal., 240 (2006), 399-428.  Google Scholar

[9]

M. J. Cáceres and Giuseppe Toscani, Kinetic approach to long time behavior of linearized fast diffusion equations, J. Stat. Phys., 128 (2007), 883-925. doi: 10.1007/s10955-007-9329-6.  Google Scholar

[10]

J. A. Carrillo, M. Di Francesco and G. Toscani, Strict contractivity of the 2-Wasserstein distance for the porous medium equation by mass-centering, Proc. Amer. Math. Soc., 135 (2007), 353-363. doi: 10.1090/S0002-9939-06-08594-7.  Google Scholar

[11]

J. A. Carrillo, A. Jüngel, 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

[12]

J. A. Carrillo, C. Lederman, P. A. Markowich and G. Toscani, Poincaré inequalities for linearizations of very fast diffusion equations, Nonlinearity, 15 (2002), 565-580. doi: 10.1088/0951-7715/15/3/303.  Google Scholar

[13]

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.  Google Scholar

[14]

D. Cordero-Erausquin, B. Nazaret and C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, Adv. Math., 182 (2004), 307-332. doi: 10.1016/S0001-8708(03)00080-X.  Google Scholar

[15]

Panagiota Daskalopoulos and Natasa Sesum, On the extinction profile of solutions to fast diffusion, J. Reine Angew. Math., 622 (2008), 95-119. doi: 10.1515/CRELLE.2008.066.  Google Scholar

[16]

Manuel Del Pino and Jean Dolbeault, Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl. (9), 81 (2002), 847-875.  Google Scholar

[17]

Jochen Denzler and Robert J. McCann, Phase transitions and symmetry breaking in singular diffusion, Proc. Natl. Acad. Sci. USA, 100 (2003), 6922-6925. doi: 10.1073/pnas.1231896100.  Google Scholar

[18]

_____, Fast diffusion to self-similarity: complete spectrum, long-time asymptotics, and numerology, Arch. Ration. Mech. Anal., 175 (2005), 301-342. doi: 10.1007/s00205-004-0336-3.  Google Scholar

[19]

Avner Friedman and Shoshana Kamin, The asymptotic behavior of gas in an $n$-dimensional porous medium, Trans. Amer. Math. Soc., 262 (1980), 551-563.  Google Scholar

[20]

Claudia Lederman and Peter A. Markowich, On fast-diffusion equations with infinite equilibrium entropy and finite equilibrium mass, Comm. Partial Differential Equations, 28 (2003), 301-332.  Google Scholar

[21]

Robert J. McCann and Dejan Slepčev, Second-order asymptotics for the fast-diffusion equation, Int. Math. Res. Not., (2006), 22 pp.  Google Scholar

[22]

William I. Newman, A Lyapunov functional for the evolution of solutions to the porous medium equation to self-similarity. I, J. Math. Phys., 25 (1984), 3120-3123. doi: 10.1063/1.526028.  Google Scholar

[23]

Felix Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.  Google Scholar

[24]

James Ralston, A Lyapunov functional for the evolution of solutions to the porous medium equation to self-similarity. II, J. Math. Phys., 25 (1984), 3124-3127. doi: 10.1063/1.526029.  Google Scholar

[25]

Giuseppe Toscani, A central limit theorem for solutions of the porous medium equation, J. Evol. Equ., 5 (2005), 185-203. doi: 10.1007/s00028-005-0183-1.  Google Scholar

[26]

Juan-Luis Vázquez, Asymptotic behaviour for the porous medium equation posed in the whole space, J. Evol. Equ., 3 (2003), 67-118. doi: 10.1007/s000280300004.  Google Scholar

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