-
Previous Article
Optimal prediction for radiative transfer: A new perspective on moment closure
- KRM Home
- This Issue
-
Next Article
Non equilibrium ionization in magnetized two-temperature thermal plasma
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 |
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
Tools
Metrics
Other articles
by authors
[Back to Top]