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 & 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.  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.   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.  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.   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.  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.  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.  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.   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.  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.  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.  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.  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.   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.  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.  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.   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.  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.  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.   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.   Google Scholar

[21]

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

[21]

Robert J. McCann and Dejan Slepčev, Second-order asymptotics for the fast-diffusion equation,, Int. Math. Res. Not., (2006).   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.  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.   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.  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.  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.  doi: 10.1007/s000280300004.  Google Scholar

[1]

Marek Fila, Michael Winkler. Sharp rate of convergence to Barenblatt profiles for a critical fast diffusion equation. Communications on Pure & 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 & Continuous Dynamical Systems - A, 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 & 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 & Continuous Dynamical Systems - A, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393

[5]

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 & Related Models, 2017, 10 (1) : 33-59. doi: 10.3934/krm.2017002

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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 & Related Models, 2017, 10 (1) : 61-91. doi: 10.3934/krm.2017003

[15]

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

[16]

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 & Continuous Dynamical Systems - A, 2016, 36 (5) : 2419-2447. doi: 10.3934/dcds.2016.36.2419

[17]

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

[18]

Kin Ming Hui, Sunghoon Kim. Existence of Neumann and singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4859-4887. doi: 10.3934/dcds.2015.35.4859

[19]

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 & Applied Analysis, 2014, 13 (6) : 2211-2228. doi: 10.3934/cpaa.2014.13.2211

[20]

Jean-Michel Roquejoffre, Luca Rossi, Violaine Roussier-Michon. Sharp large time behaviour in $ N $-dimensional Fisher-KPP equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7265-7290. doi: 10.3934/dcds.2019303

2018 Impact Factor: 1.38

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]