American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Robust exponential attractors for population dynamics models with infinite time delay
  • DCDS-B Home
  • This Issue
  • Next Article
    Modelling and long-time behaviour for phase transitions with entropy balance and thermal memory conductivity
September  2006, 6(5): 1027-1050. doi: 10.3934/dcdsb.2006.6.1027

Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations

José A. Carrillo 1, , Jean Dolbeault 2, , Ivan Gentil 3, and  Ansgar Jüngel 4,

1. 

ICREA-Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain
2. 

Ceremade (UMR CNRS 7534), Université Paris Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris, Cédex 16, France
3. 

Ceremade, UMR CNRS 7534, Université Paris-Dauphine, Place du Maréchal De Lattre de Tassigny, 75775 PARIS Cedex 16, France
4. 

Fachbereich Physik, Mathematik und Informatik, Universität Mainz, Staudingerweg 9, 55099 Mainz, Germany

Received  September 2005 Revised  March 2006 Published  June 2006

In this paper, we prove new functional inequalities of Poincaré type on the one-dimensional torus $S^1$ and explore their implications for the long-time asymptotics of periodic solutions of nonlinear singular or degenerate parabolic equations of second and fourth order. We generically prove a global algebraic decay of an entropy functional, faster than exponential for short times, and an asymptotically exponential convergence of positive solutions towards their average. The asymptotically exponential regime is valid for a larger range of parameters for all relevant cases of application: porous medium/fast diffusion, thin film and logarithmic fourth order nonlinear diffusion equations. The techniques are inspired by direct entropy-entropy production methods and based on appropriate Poincaré type inequalities.
Keywords: Poincare inequality, higher-order nonlinear PDEs, entropy production, entropy-entropy production method, Sobolev estimates, entropy, fast diffusion equation, parabolic equations, Logarithmic Sobolev inequality, long-time behavior, thin film equation., porous media equation.
Mathematics Subject Classification: MSC (2000): 35K35, 35K55, 35G25, 35B40, 26D10,35B45, 76A20, 76D08, 76S0.
Citation: José A. Carrillo, Jean Dolbeault, Ivan Gentil, Ansgar Jüngel. Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1027-1050. doi: 10.3934/dcdsb.2006.6.1027
[1]

Seok-Bae Yun. Entropy production for ellipsoidal BGK model of the Boltzmann equation. Kinetic & Related Models, 2016, 9 (3) : 605-619. doi: 10.3934/krm.2016009
[2]

Amit Einav. On Villani's conjecture concerning entropy production for the Kac Master equation. Kinetic & Related Models, 2011, 4 (2) : 479-497. doi: 10.3934/krm.2011.4.479
[3]

Ansgar Jüngel, Ingrid Violet. Mixed entropy estimates for the porous-medium equation with convection. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 783-796. doi: 10.3934/dcdsb.2009.12.783
[4]

Timothy Chumley, Renato Feres. Entropy production in random billiards. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1319-1346. doi: 10.3934/dcds.2020319
[5]

Eric A. Carlen, Maria C. Carvalho, Amit Einav. Entropy production inequalities for the Kac Walk. Kinetic & Related Models, 2018, 11 (2) : 219-238. doi: 10.3934/krm.2018012
[6]

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
[7]

Yue-Jun Peng, Yong-Fu Yang. Long-time behavior and stability of entropy solutions for linearly degenerate hyperbolic systems of rich type. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3683-3706. doi: 10.3934/dcds.2015.35.3683
[8]

Tong Li, Kun Zhao. Global existence and long-time behavior of entropy weak solutions to a quasilinear hyperbolic blood flow model. Networks & Heterogeneous Media, 2011, 6 (4) : 625-646. doi: 10.3934/nhm.2011.6.625
[9]

Swann Marx, Tillmann Weisser, Didier Henrion, Jean Bernard Lasserre. A moment approach for entropy solutions to nonlinear hyperbolic PDEs. Mathematical Control & Related Fields, 2020, 10 (1) : 113-140. doi: 10.3934/mcrf.2019032
[10]

Peter V. Gordon, Cyrill B. Muratov. Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source. Networks & Heterogeneous Media, 2012, 7 (4) : 767-780. doi: 10.3934/nhm.2012.7.767
[11]

Brahim Alouini. Long-time behavior of a Bose-Einstein equation in a two-dimensional thin domain. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1629-1643. doi: 10.3934/cpaa.2011.10.1629
[12]

Elena Bonetti, Pierluigi Colli, Mauro Fabrizio, Gianni Gilardi. Modelling and long-time behaviour for phase transitions with entropy balance and thermal memory conductivity. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1001-1026. doi: 10.3934/dcdsb.2006.6.1001
[13]

Andrea Giorgini. On the Swift-Hohenberg equation with slow and fast dynamics: well-posedness and long-time behavior. Communications on Pure & Applied Analysis, 2016, 15 (1) : 219-241. doi: 10.3934/cpaa.2016.15.219
[14]

Felix X.-F. Ye, Hong Qian. Stochastic dynamics Ⅱ: Finite random dynamical systems, linear representation, and entropy production. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4341-4366. doi: 10.3934/dcdsb.2019122
[15]

Marcel Braukhoff, Ansgar Jüngel. Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3335-3355. doi: 10.3934/dcdsb.2020234
[16]

John Kieffer and En-hui Yang. Ergodic behavior of graph entropy. Electronic Research Announcements, 1997, 3: 11-16.

[17]

Kenneth Hvistendahl Karlsen, Nils Henrik Risebro. On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1081-1104. doi: 10.3934/dcds.2003.9.1081
[18]

Giuseppe Maria Coclite, Lorenzo Di Ruvo. A note on the convergence of the solution of the high order Camassa-Holm equation to the entropy ones of a scalar conservation law. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1247-1282. doi: 10.3934/dcds.2017052
[19]

Yu-Zhao Wang. $ \mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2441-2454. doi: 10.3934/cpaa.2018116
[20]

Milena Stanislavova, Atanas Stefanov. Effective estimates of the higher Sobolev norms for the Kuramoto-Sivashinsky equation. Conference Publications, 2009, 2009 (Special) : 729-738. doi: 10.3934/proc.2009.2009.729

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences