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

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

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.
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.

[1]

Marek Fila, Michael Winkler. Sharp rate of convergence to Barenblatt profiles for a critical fast diffusion equation. Communications on Pure and Applied Analysis, 2015, 14 (1) : 107-119. doi: 10.3934/cpaa.2015.14.107

[2]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[3]

Judith Vancostenoble. Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 761-790. doi: 10.3934/dcdss.2011.4.761

[4]

Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393

[5]

Kin Ming Hui, Jinwan Park. Asymptotic behaviour of singular solution of the fast diffusion equation in the punctured euclidean space. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5473-5508. doi: 10.3934/dcds.2021085

[6]

Gabriele Grillo, Matteo Muratori, Fabio Punzo. On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5927-5962. doi: 10.3934/dcds.2015.35.5927

[7]

Matteo Bonforte, Jean Dolbeault, Matteo Muratori, Bruno Nazaret. Weighted fast diffusion equations (Part Ⅰ): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities. Kinetic and Related Models, 2017, 10 (1) : 33-59. doi: 10.3934/krm.2017002

[8]

Liviu I. Ignat, Ademir F. Pazoto. Large time behaviour for a nonlocal diffusion - convection equation related with gas dynamics. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3575-3589. doi: 10.3934/dcds.2014.34.3575

[9]

Marek Fila, Hannes Stuke. Special asymptotics for a critical fast diffusion equation. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 725-735. doi: 10.3934/dcdss.2014.7.725

[10]

Gabriele Grillo, Matteo Muratori, Maria Michaela Porzio. Porous media equations with two weights: Smoothing and decay properties of energy solutions via Poincaré inequalities. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3599-3640. doi: 10.3934/dcds.2013.33.3599

[11]

Marek Fila, Juan-Luis Vázquez, Michael Winkler. A continuum of extinction rates for the fast diffusion equation. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1129-1147. doi: 10.3934/cpaa.2011.10.1129

[12]

Gianluca Mola. Recovering a large number of diffusion constants in a parabolic equation from energy measurements. Inverse Problems and Imaging, 2018, 12 (3) : 527-543. doi: 10.3934/ipi.2018023

[13]

Vladimir Varlamov. Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 675-702. doi: 10.3934/dcds.2001.7.675

[14]

Roberta Bosi, Jean Dolbeault, Maria J. Esteban. Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators. Communications on Pure and Applied Analysis, 2008, 7 (3) : 533-562. doi: 10.3934/cpaa.2008.7.533

[15]

Matteo Bonforte, Jean Dolbeault, Matteo Muratori, Bruno Nazaret. Weighted fast diffusion equations (Part Ⅱ): Sharp asymptotic rates of convergence in relative error by entropy methods. Kinetic and Related Models, 2017, 10 (1) : 61-91. doi: 10.3934/krm.2017003

[16]

Shu-Yu Hsu. Super fast vanishing solutions of the fast diffusion equation. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5383-5414. doi: 10.3934/dcds.2020232

[17]

Shifeng Geng, Lina Zhang. Large-time behavior of solutions for the system of compressible adiabatic flow through porous media with nonlinear damping. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2211-2228. doi: 10.3934/cpaa.2014.13.2211

[18]

Jerry L. Bona, Laihan Luo. Large-time asymptotics of the generalized Benjamin-Ono-Burgers equation. Discrete and Continuous Dynamical Systems - S, 2011, 4 (1) : 15-50. doi: 10.3934/dcdss.2011.4.15

[19]

Marcello D'Abbicco, Ruy Coimbra Charão, Cleverson Roberto da Luz. Sharp time decay rates on a hyperbolic plate model under effects of an intermediate damping with a time-dependent coefficient. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2419-2447. doi: 10.3934/dcds.2016.36.2419

[20]

H. T. Liu. Impulsive effects on the existence of solutions for a fast diffusion equation. Conference Publications, 2001, 2001 (Special) : 248-253. doi: 10.3934/proc.2001.2001.248

2020 Impact Factor: 1.432

Metrics

  • PDF downloads (79)
  • HTML views (0)
  • Cited by (14)

Other articles
by authors

[Back to Top]