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

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[21]	Robert J. McCann and Dejan Slepčev, Second-order asymptotics for the fast-diffusion equation, Int. Math. Res. Not., (2006), 22 pp.
[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.
[23]	Felix Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[21]	Robert J. McCann and Dejan Slepčev, Second-order asymptotics for the fast-diffusion equation, Int. Math. Res. Not., (2006), 22 pp.
[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.
[23]	Felix Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.
[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.
[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.
[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.

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