1,796
Views
228
CrossRef citations to date
0
Altmetric
Original Articles

ON CONVEX SOBOLEV INEQUALITIES AND THE RATE OF CONVERGENCE TO EQUILIBRIUM FOR FOKKER-PLANCK TYPE EQUATIONS

, , &
Pages 43-100 | Received 01 Jan 1999, Published online: 07 Feb 2007

Keep up to date with the latest research on this topic with citation updates for this article.

Read on this site (5)

Bin Li & Jie-qiong Shen. (2017) The Wigner(–Poisson)–Fokker–Planck equation with singular potential. Applicable Analysis 96:4, pages 563-577.
Read now
Simone Calogero. (2012) Exponential Convergence to Equilibrium for Kinetic Fokker-Planck Equations. Communications in Partial Differential Equations 37:8, pages 1357-1390.
Read now
JoséA. Carrillo & Thierry Goudon. (2006) Stability and Asymptotic Analysis of a Fluid-Particle Interaction Model. Communications in Partial Differential Equations 31:9, pages 1349-1379.
Read now
Maria J. Cáceres, José A. Carrillo & Thierry Goudon. (2003) Equilibration Rate for the Linear Inhomogeneous Relaxation-Time Boltzmann Equation for Charged Particles. Communications in Partial Differential Equations 28:5-6, pages 969-989.
Read now
Claudia Lederman & Peter A. Markowich. (2003) On Fast-Diffusion Equations with Infinite Equilibrium Entropy and Finite Equilibrium Mass. Communications in Partial Differential Equations 28:1-2, pages 301-332.
Read now

Articles from other publishers (223)

Erhan Bayraktar, Qi Feng & Wuchen Li. (2023) Exponential Entropy Dissipation for Weakly Self-Consistent Vlasov–Fokker–Planck Equations. Journal of Nonlinear Science 34:1.
Crossref
Alessio Porretta. (2024) Decay rates of convergence for Fokker-Planck equations with confining drift. Advances in Mathematics 436, pages 109393.
Crossref
Chen Liu, Yuan Gao & Xiangxiong Zhang. (2023) Structure Preserving Schemes for Fokker–Planck Equations of Irreversible Processes. Journal of Scientific Computing 98:1.
Crossref
Bowen Li & Jianfeng Lu. (2023) Interpolation Between Modified Logarithmic Sobolev and Poincaré Inequalities for Quantum Markovian Dynamics. Journal of Statistical Physics 190:10.
Crossref
Wuchen Li, Siting Liu & Stanley Osher. (2023) Controlling conservation laws I: Entropy–entropy flux. Journal of Computational Physics 480, pages 112019.
Crossref
Henri Berestycki, Benoît Desjardins, Joshua S. Weitz & Jean-Marc Oury. (2023) Epidemic modeling with heterogeneity and social diffusion. Journal of Mathematical Biology 86:4.
Crossref
José A. Carrillo, Rishabh S. Gvalani & Jeremy S.-H. Wu. (2023) An invariance principle for gradient flows in the space of probability measures. Journal of Differential Equations 345, pages 233-284.
Crossref
José A. Carrillo, David Gómez-Castro, Yao Yao & Chongchun Zeng. (2023) Asymptotic Simplification of Aggregation-Diffusion Equations Towards the Heat kernel. Archive for Rational Mechanics and Analysis 247:1.
Crossref
Chaoen Zhang. (2023) Hypocoercivity and global hypoellipticity for the kinetic Fokker-Planck equation in $ H^k $ spaces. Kinetic and Related Models 0:0, pages 0-0.
Crossref
Yekaterina Epshteyn, Chang Liu, Chun Liu & Masashi Mizuno. (2022) Nonlinear inhomogeneous Fokker–Planck models: Energetic-variational structures and long-time behavior. Analysis and Applications 20:06, pages 1295-1356.
Crossref
Yekaterina Epshteyn, Chun Liu & Masashi Mizuno. (2022) A stochastic model of grain boundary dynamics: A Fokker–Planck perspective. Mathematical Models and Methods in Applied Sciences 32:11, pages 2189-2236.
Crossref
Louis Labuschagne & W. Adam Majewski. (2021) Quantum Fokker–Planck Dynamics. Annales Henri Poincaré 23:5, pages 1659-1691.
Crossref
Marcel Braukhoff, Amit Einav & Bao Quoc Tang. (2022) Quantitative dynamics of irreversible enzyme reaction–diffusion systems * . Nonlinearity 35:4, pages 1876-1926.
Crossref
Giovanni Conforti. (2022) A probabilistic approach to convex (ϕ)-entropy decay for Markov chains. The Annals of Applied Probability 32:2.
Crossref
Marcel Braukhoff, Ilaria Perugia & Paul Stocker. (2022) An Entropy Structure Preserving Space-Time Formulation for Cross-Diffusion Systems: Analysis and Galerkin Discretization. SIAM Journal on Numerical Analysis 60:1, pages 364-395.
Crossref
Anton Arnold & Beatrice Signorello. (2022) Optimal non-symmetric Fokker-Planck equation for the convergence to a given equilibrium. Kinetic and Related Models 15:5, pages 753.
Crossref
Quentin Chauleur. (2022) The isothermal limit for the compressible Euler equations with damping. Discrete and Continuous Dynamical Systems - B 27:12, pages 7671.
Crossref
Jose A. Carrillo, David Gómez-Castro & Juan Luis Vázquez. (2022) Infinite-time concentration in aggregation–diffusion equations with a given potential. Journal de Mathématiques Pures et Appliquées 157, pages 346-398.
Crossref
Jean Dolbeault. (2021) Functional Inequalities: Nonlinear Flows and Entropy Methods as a Tool for Obtaining Sharp and Constructive Results. Milan Journal of Mathematics 89:2, pages 355-386.
Crossref
Jean Dolbeault & Xingyu Li. (2021) Generalized Logarithmic Hardy–Littlewood–Sobolev Inequality. International Mathematics Research Notices 2021:23, pages 17862-17874.
Crossref
Mattia Zanella, Chiara Bardelli, Giacomo Dimarco, Silvia Deandrea, Pietro Perotti, Mara Azzi, Silvia Figini & Giuseppe Toscani. (2021) A data-driven epidemic model with social structure for understanding the COVID-19 infection on a heavily affected Italian province. Mathematical Models and Methods in Applied Sciences 31:12, pages 2533-2570.
Crossref
Muka Nakazato & Sosuke Ito. (2021) Geometrical aspects of entropy production in stochastic thermodynamics based on Wasserstein distance. Physical Review Research 3:4.
Crossref
Jacob Bedrossian & Kyle Liss. (2021) Quantitative spectral gaps for hypoelliptic stochastic differential equations with small noise. Probability and Mathematical Physics 2:3, pages 477-532.
Crossref
Jean-Luc Thiffeault. (2021) Nonuniform mixing. Physical Review Fluids 6:9.
Crossref
Frederic Weber & Rico Zacher. (2021) The entropy method under curvature-dimension conditions in the spirit of Bakry-Émery in the discrete setting of Markov chains. Journal of Functional Analysis 281:5, pages 109061.
Crossref
El Houssaine Quenjel. (2021) Analysis of Accurate and Stable Nonlinear Finite Volume Scheme for Anisotropic Diffusion Equations with Drift on Simplicial Meshes. Journal of Scientific Computing 88:3.
Crossref
Lanoir Addala, Jean Dolbeault, Xingyu Li & M. Lazhar Tayeb. (2021) $$\text {L}^2$$-Hypocoercivity and Large Time Asymptotics of the Linearized Vlasov–Poisson–Fokker–Planck System. Journal of Statistical Physics 184:1.
Crossref
Quentin Chauleur. (2021) Dynamics of the Schrödinger–Langevin equation. Nonlinearity 34:4, pages 1943-1974.
Crossref
Wei Zhang. (2021) Some new results on relative entropy production, time reversal, and optimal control of time-inhomogeneous diffusion processes. Journal of Mathematical Physics 62:4.
Crossref
Guillaume Ferriere. (2021) Convergence rate in Wasserstein distance and semiclassical limit for the defocusing logarithmic Schrödinger equation. Analysis & PDE 14:2, pages 617-666.
Crossref
Thierry Goudon. (2021) Diffusion regime with a high and oscillating field. Tunisian Journal of Mathematics 3:2, pages 395-445.
Crossref
Josephine Evans. (2021) Hypocoercivity in Phi-Entropy for the Linear Relaxation Boltzmann Equation on the Torus. SIAM Journal on Mathematical Analysis 53:2, pages 1357-1378.
Crossref
Shui-Nee Chow, Jun Lu & Hao-Min Zhou. 2021. Mathematical Methods in Image Processing and Inverse Problems. Mathematical Methods in Image Processing and Inverse Problems 181 196 .
Claire Chainais-Hillairet & Maxime Herda. (2020) Large-time behaviour of a family of finite volume schemes for boundary-driven convection–diffusion equations. IMA Journal of Numerical Analysis 40:4, pages 2473-2504.
Crossref
Luca Alasio, Helene Ranetbauer, Markus Schmidtchen & Marie-Therese Wolfram. (2020) Trend to equilibrium for systems with small cross-diffusion. ESAIM: Mathematical Modelling and Numerical Analysis 54:5, pages 1661-1688.
Crossref
I. Honoré, S. Menozzi & G. Pagès. (2020) Non-asymptotic Gaussian estimates for the recursive approximation of the invariant distribution of a diffusion. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 56:3.
Crossref
Yanmin Mu & Dehua Wang. (2020) Global well-posedness and optimal large-time behavior of strong solutions to the non-isentropic particle-fluid flows. Calculus of Variations and Partial Differential Equations 59:4.
Crossref
Hao Wang. (2020) Global existence and decay of solutions for soft potentials to the Fokker–Planck–Boltzmann equation without cut-off. Journal of Mathematical Analysis and Applications 486:2, pages 123947.
Crossref
Filomena Feo, Yanghong Huang & Bruno Volzone. (2019) Long-time asymptotics for a 1D nonlocal porous medium equation with absorption or convection. Communications in Contemporary Mathematics 22:03, pages 1950015.
Crossref
Shin-ichi Ohta & Asuka Takatsu. (2019) Equality in the logarithmic Sobolev inequality. manuscripta mathematica 162:1-2, pages 271-282.
Crossref
Giulia Furioli, Ada Pulvirenti, Elide Terraneo & Giuseppe Toscani. (2020) Non-Maxwellian kinetic equations modeling the dynamics of wealth distribution. Mathematical Models and Methods in Applied Sciences 30:04, pages 685-725.
Crossref
Shuangqian Liu & Xuan Ma. (2020) The relativistic Vlasov–Maxwell–Fokker–Planck system in the whole space. Nonlinearity 33:4, pages 1789-1811.
Crossref
Vincent Calvez, Thomas Lepoutre, Nicolas Meunier & Nicolas Muller. (2020) Non-linear analysis of a model for yeast cell communication. ESAIM: Mathematical Modelling and Numerical Analysis 54:2, pages 619-648.
Crossref
Stanislav Kondratyev & Dmitry Vorotnikov. (2020) Convex Sobolev inequalities related to unbalanced optimal transport. Journal of Differential Equations 268:7, pages 3705-3724.
Crossref
Clément Cancès, Claire Chainais-Hillairet, Maxime Herda & Stella Krell. (2020) Large Time Behavior of Nonlinear Finite Volume Schemes for Convection-Diffusion Equations. SIAM Journal on Numerical Analysis 58:5, pages 2544-2571.
Crossref
Alfredo Garbuno-Inigo, Franca Hoffmann, Wuchen Li & Andrew M. Stuart. (2020) Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler. SIAM Journal on Applied Dynamical Systems 19:1, pages 412-441.
Crossref
M Burger, E Resmerita & M Benning. (2019) An entropic Landweber method for linear ill-posed problems. Inverse Problems 36:1, pages 015009.
Crossref
Anton Arnold, Shi Jin & Tobias Wöhrer. (2020) Sharp decay estimates in local sensitivity analysis for evolution equations with uncertainties: From ODEs to linear kinetic equations. Journal of Differential Equations 268:3, pages 1156-1204.
Crossref
Djalil Chafaï & Joseph Lehec. 2020. Geometric Aspects of Functional Analysis. Geometric Aspects of Functional Analysis 219 246 .
Till D. Frank. 2020. Synergetics. Synergetics 149 182 .
Giulia Furioli, Ada Pulvirenti, Elide Terraneo & Giuseppe Toscani. (2019) Wright–Fisher–type equations for opinion formation, large time behavior and weighted logarithmic-Sobolev inequalities. Annales de l'Institut Henri Poincaré C, Analyse non linéaire 36:7, pages 2065-2082.
Crossref
Wuchen Li & Lexing Ying. (2019) Hessian transport gradient flows. Research in the Mathematical Sciences 6:4.
Crossref
Maxime Breden & Christian Kuehn. (2019) Rigorous validation of stochastic transition paths. Journal de Mathématiques Pures et Appliquées 131, pages 88-129.
Crossref
Youshan Tao & Michael Winkler. (2019) Large time behavior in a forager–exploiter model with different taxis strategies for two groups in search of food. Mathematical Models and Methods in Applied Sciences 29:11, pages 2151-2182.
Crossref
Yu Cao, Jianfeng Lu & Yulong Lu. (2019) Exponential Decay of Rényi Divergence Under Fokker–Planck Equations. Journal of Statistical Physics 176:5, pages 1172-1184.
Crossref
Xuan Ma. (2019) On the Cauchy problem for the relativistic Vlasov-Poisson-Fokker-Planck system. Journal of Mathematical Analysis and Applications 475:2, pages 1044-1053.
Crossref
Vladimir I. Bogachev, Michael Röckner & Stanislav V. Shaposhnikov. (2019) Convergence in variation of solutions of nonlinear Fokker–Planck–Kolmogorov equations to stationary measures. Journal of Functional Analysis 276:12, pages 3681-3713.
Crossref
Youshan Tao & Michael Winkler. (2019) Global smooth solvability of a parabolic–elliptic nutrient taxis system in domains of arbitrary dimension. Journal of Differential Equations 267:1, pages 388-406.
Crossref
Rémi Carles, Kleber Carrapatoso & Matthieu Hillairet. (2019) Rigidity results in generalized isothermal fluids. Annales Henri Lebesgue 1, pages 47-85.
Crossref
Michael Winkler. (2019) Global generalized solutions to a multi-dimensional doubly tactic resource consumption model accounting for social interactions. Mathematical Models and Methods in Applied Sciences 29:03, pages 373-418.
Crossref
Jukka Kemppainen & Rico Zacher. (2019) Long-time behavior of non-local in time Fokker–Planck equations via the entropy method. Mathematical Models and Methods in Applied Sciences 29:02, pages 209-235.
Crossref
Franz Achleitner, Ansgar Jüngel & Masakazu Yamamoto. (2019) Large-time asymptotics of a fractional drift–diffusion–Poisson system via the entropy method. Nonlinear Analysis 179, pages 270-293.
Crossref
Karsten Matthies & Florian Theil. (2019) Rescaled Objective Solutions of Fokker--Planck and Boltzmann equations. SIAM Journal on Mathematical Analysis 51:2, pages 1321-1348.
Crossref
Lara Neureither & Carsten Hartmann. 2019. Stochastic Dynamics Out of Equilibrium. Stochastic Dynamics Out of Equilibrium 391 410 .
Jean Dolbeault & Xingyu Li. (2018) φ-Entropies: convexity, coercivity and hypocoercivity for Fokker–Planck and kinetic Fokker–Planck equations. Mathematical Models and Methods in Applied Sciences 28:13, pages 2637-2666.
Crossref
Thibaut Demaerel, Christian Maes & Karel Netočný. (2018) Stabilization in the Eye of a Cyclone. Annales Henri Poincaré 19:9, pages 2673-2699.
Crossref
Katarína Boďová, Jan Haskovec & Peter Markowich. (2018) Well posedness and maximum entropy approximation for the dynamics of quantitative traits. Physica D: Nonlinear Phenomena 376-377, pages 108-120.
Crossref
Clément Cancès, Claire Chainais-Hillairet & Stella Krell. (2018) Numerical Analysis of a Nonlinear Free-Energy Diminishing Discrete Duality Finite Volume Scheme for Convection Diffusion Equations. Computational Methods in Applied Mathematics 18:3, pages 407-432.
Crossref
Rémi Carles & Isabelle Gallagher. (2018) Universal dynamics for the defocusing logarithmic Schrödinger equation. Duke Mathematical Journal 167:9.
Crossref
Anton Arnold, Amit Einav & Tobias Wöhrer. (2018) On the rates of decay to equilibrium in degenerate and defective Fokker–Planck equations. Journal of Differential Equations 264:11, pages 6843-6872.
Crossref
Klemens Fellner & Bao Quoc Tang. (2018) Convergence to equilibrium of renormalised solutions to nonlinear chemical reaction–diffusion systems. Zeitschrift für angewandte Mathematik und Physik 69:3.
Crossref
Jan Haskovec, Sabine Hittmeir, Peter Markowich & Alexander Mielke. (2018) Decay to Equilibrium for Energy-Reaction-Diffusion Systems. SIAM Journal on Mathematical Analysis 50:1, pages 1037-1075.
Crossref
Thierry Goudon, Arthur Vavasseur & Ricardo Alonso. (2017) Damping of particles interacting with a vibrating medium. Annales de l'Institut Henri Poincaré C, Analyse non linéaire 34:7, pages 1727-1758.
Crossref
Francis Filbet & Maxime Herda. (2017) A finite volume scheme for boundary-driven convection–diffusion equations with relative entropy structure. Numerische Mathematik 137:3, pages 535-577.
Crossref
Gheorghe Nenciu & Irina Nenciu. (2017) Drift-diffusion equations on domains in : Essential self-adjointness and stochastic completeness . Journal of Functional Analysis 273:8, pages 2619-2654.
Crossref
Ansgar Jüngel & Wen Yue. (2017) Discrete Beckner inequalities via the Bochner–Bakry–Emery approach for Markov chains. The Annals of Applied Probability 27:4.
Crossref
Klemens Fellner & Bao Quoc Tang. (2017) Explicit exponential convergence to equilibrium for nonlinear reaction–diffusion systems with detailed balance condition. Nonlinear Analysis 159, pages 145-180.
Crossref
Simon Eberle, Barbara Niethammer & André Schlichting. (2017) Gradient flow formulation and longtime behaviour of a constrained Fokker–Planck equation. Nonlinear Analysis 158, pages 142-167.
Crossref
Bernard Chazelle, Quansen Jiu, Qianxiao Li & Chu Wang. (2017) Well-posedness of the limiting equation of a noisy consensus model in opinion dynamics. Journal of Differential Equations 263:1, pages 365-397.
Crossref
Kleber Carrapatoso, Laurent Desvillettes & Lingbing He. (2017) Estimates for the Large Time Behavior of the Landau Equation in the Coulomb Case. Archive for Rational Mechanics and Analysis 224:2, pages 381-420.
Crossref
ANSGAR JÜNGEL, CHRISTIAN KUEHN & LARA TRUSSARDI. (2016) A meeting point of entropy and bifurcations in cross-diffusion herding. European Journal of Applied Mathematics 28:2, pages 317-356.
Crossref
Ansgar Jüngel & Nicola Zamponi. (2017) A cross-diffusion system derived from a Fokker–Planck equation with partial averaging. Zeitschrift für angewandte Mathematik und Physik 68:1.
Crossref
Marianne Bessemoulin-Chatard & Claire Chainais-Hillairet. (2017) Exponential decay of a finite volume scheme to the thermal equilibrium for drift–diffusion systems. Journal of Numerical Mathematics 25:3.
Crossref
Klemens Fellner, Wolfang Prager & Bao Q. Tang. (2017) The entropy method for reaction-diffusion systems without detailed balance: First order chemical reaction networks. Kinetic & Related Models 10:4, pages 1055-1087.
Crossref
Matteo Bonforte, Jean Dolbeault, Matteo Muratori & Bruno Nazaret. (2017) Weighted fast diffusion equations (Part Ⅰ): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities. Kinetic & Related Models 10:1, pages 33-59.
Crossref
Giulia Furioli, Ada Pulvirenti, Elide Terraneo & Giuseppe Toscani. (2017) Fokker–Planck equations in the modeling of socio-economic phenomena. Mathematical Models and Methods in Applied Sciences 27:01, pages 115-158.
Crossref
Laurent Desvillettes, Klemens Fellner & Bao Quoc Tang. (2017) Trend to Equilibrium for Reaction-Diffusion Systems Arising from Complex Balanced Chemical Reaction Networks. SIAM Journal on Mathematical Analysis 49:4, pages 2666-2709.
Crossref
Till D. Frank. 2017. Encyclopedia of Complexity and Systems Science. Encyclopedia of Complexity and Systems Science 1 36 .
José Antonio Carrillo, Manuel del Pino, Alessio Figalli, Giuseppe Mingione & Juan Luis VázquezJuan Luis Vázquez. 2017. Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions. Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions 205 278 .
Hiroaki Yoshida. (2016) A Dissipation of Relative Entropy by Diffusion Flows. Entropy 19:1, pages 9.
Crossref
Jean Dolbeault & An Zhang. (2016) Optimal Functional Inequalities for Fractional Operators on the Sphere and Applications. Advanced Nonlinear Studies 16:4, pages 863-880.
Crossref
V. Bonnaillie-Noël, J. A. Carrillo, T. Goudon & G. A. Pavliotis. (2016) Efficient numerical calculation of drift and diffusion coefficients in the diffusion approximation of kinetic equations. IMA Journal of Numerical Analysis 36:4, pages 1536-1569.
Crossref
Giuseppe Toscani. (2016) Entropy Inequalities for Stable Densities and Strengthened Central Limit Theorems. Journal of Statistical Physics 165:2, pages 371-389.
Crossref
V.I. Bogachev, M. Röckner & S.V. Shaposhnikov. (2016) Distances between transition probabilities of diffusions and applications to nonlinear Fokker–Planck–Kolmogorov equations. Journal of Functional Analysis 271:5, pages 1262-1300.
Crossref
Mario Bukal. (2016) A family of fourth-order q-logarithmic equations. Journal of Mathematical Analysis and Applications 438:1, pages 142-161.
Crossref
Zhengrong Liu & Hao Tang. (2016) Global well-posedness for the Fokker–Planck–Boltzmann equation in Besov–Chemin–Lerner type spaces. Journal of Differential Equations 260:12, pages 8638-8674.
Crossref
Tony Lelièvre & Gabriel Stoltz. (2016) Partial differential equations and stochastic methods in molecular dynamics. Acta Numerica 25, pages 681-880.
Crossref
Joaquin Fontbona & Benjamin Jourdain. (2016) A trajectorial interpretation of the dissipations of entropy and Fisher information for stochastic differential equations. The Annals of Probability 44:1.
Crossref
Ansgar JüngelAnsgar Jüngel. 2016. Entropy Methods for Diffusive Partial Differential Equations. Entropy Methods for Diffusive Partial Differential Equations 19 44 .
Ansgar JüngelAnsgar Jüngel. 2016. Entropy Methods for Diffusive Partial Differential Equations. Entropy Methods for Diffusive Partial Differential Equations 1 17 .
Franz Achleitner, Anton Arnold & Eric A. Carlen. 2016. From Particle Systems to Partial Differential Equations III. From Particle Systems to Partial Differential Equations III 1 37 .
Martin Burger. 2016. Advances in Mathematical Modeling, Optimization and Optimal Control. Advances in Mathematical Modeling, Optimization and Optimal Control 3 33 .
Giuseppe Toscani. (2015) A Strengthened Entropy Power Inequality for Log-Concave Densities. IEEE Transactions on Information Theory 61:12, pages 6550-6559.
Crossref
Tat Dat Tran, Julian Hofrichter & Jürgen Jost. (2015) The free energy method and the Wright–Fisher model with 2 alleles. Theory in Biosciences 134:3-4, pages 83-92.
Crossref
Alexander Mielke, Jan Haskovec & Peter A. Markowich. (2014) On Uniform Decay of the Entropy for Reaction–Diffusion Systems. Journal of Dynamics and Differential Equations 27:3-4, pages 897-928.
Crossref
Jonathan Zinsl. (2015) Exponential convergence to equilibrium in a Poisson-Nernst-Planck-type system with nonlinear diffusion. Discrete and Continuous Dynamical Systems 36:5, pages 2915-2930.
Crossref
J. A. Carrillo, S. Mancini & M.-B. Tran. (2017) On the Exponential Convergence Rate for a Non-Gradient Fokker-Planck Equation in Computational Neuroscience. Journal of Elliptic and Parabolic Equations 1:2, pages 271-279.
Crossref
Jean Dolbeault, Maria J. Esteban & Gaspard Jankowiak. (2015) The Moser-Trudinger-Onofri inequality. Chinese Annals of Mathematics, Series B 36:5, pages 777-802.
Crossref
Marzia Bisi, José A. Cañizo & Bertrand Lods. (2015) Entropy dissipation estimates for the linear Boltzmann operator. Journal of Functional Analysis 269:4, pages 1028-1069.
Crossref
Christian Schmeiser, Gaël Raoul & Vincent Calvez. (2015) Confinement by biased velocity jumps: Aggregation of <em> escherichia coli</em>. Kinetic and Related Models 8:4, pages 651-666.
Crossref
Hailiang Liu & Hui Yu. (2014) The Entropy Satisfying Discontinuous Galerkin Method for Fokker–Planck equations. Journal of Scientific Computing 62:3, pages 803-830.
Crossref
J.A. Carrillo, Y. Huang, M.C. Santos & J.L. Vázquez. (2015) Exponential convergence towards stationary states for the 1D porous medium equation with fractional pressure. Journal of Differential Equations 258:3, pages 736-763.
Crossref
Lan LUO & Xinping ZHANG. (2015) Global classical solutions for quantum kinetic fokker-planck equations. Acta Mathematica Scientia 35:1, pages 140-156.
Crossref
J A Carrillo & G Toscani. (2014) Rényi entropy and improved equilibration rates to self-similarity for nonlinear diffusion equations. Nonlinearity 27:12, pages 3159-3177.
Crossref
Jacob Bedrossian & Nader Masmoudi. (2014) Existence, Uniqueness and Lipschitz Dependence for Patlak–Keller–Segel and Navier–Stokes in $${\mathbb{R}^2}$$ R 2 with Measure-Valued Initial Data. Archive for Rational Mechanics and Analysis 214:3, pages 717-801.
Crossref
Martin Grothaus & Patrik Stilgenbauer. (2014) Hypocoercivity for Kolmogorov backward evolution equations and applications. Journal of Functional Analysis 267:10, pages 3515-3556.
Crossref
Aya Nemoto & Hiroaki Yoshida. (2014) The free logarithmic Sobolev and the free transportation cost inequalities by time integrations. Infinite Dimensional Analysis, Quantum Probability and Related Topics 17:03, pages 1450022.
Crossref
Thierry Bodineau, Joel Lebowitz, Clément Mouhot & Cédric Villani. (2014) Lyapunov functionals for boundary-driven nonlinear drift–diffusion equations. Nonlinearity 27:9, pages 2111-2132.
Crossref
Jean Dolbeault, Maria J. Esteban & Michael Loss. (2014) Nonlinear flows and rigidity results on compact manifolds. Journal of Functional Analysis 267:5, pages 1338-1363.
Crossref
Ge Yang. (2014) Global well-posedness for the diffusion equation of population genetics. Journal of Mathematical Analysis and Applications 415:2, pages 760-778.
Crossref
Yuanjie Lei, Linjie Xiong & Huijiang Zhao. (2014) One-species Vlasov-Poisson-Landau system near Maxwellians in the whole space. Kinetic & Related Models 7:3, pages 551-590.
Crossref
Linjie Xiong, Tao Wang & Lusheng Wang. (2014) Global existence and decay of solutions to the Fokker-Planck-Boltzmann equation. Kinetic & Related Models 7:1, pages 169-194.
Crossref
Jean Dolbeault, Maria J. Esteban, Michał Kowalczyk & Michael Loss. (2014) Improved interpolation inequalities on the sphere. Discrete & Continuous Dynamical Systems - S 7:4, pages 695-724.
Crossref
Laurent Desvillettes & Klemens Fellner. 2014. System Modeling and Optimization. System Modeling and Optimization 96 104 .
Jean Dolbeault, Maria J. Esteban, Michal Kowalczyk & Michael Loss. 2014. Partial Differential Equations: Theory, Control and Approximation. Partial Differential Equations: Theory, Control and Approximation 225 242 .
Dominique Bakry, Ivan Gentil & Michel LedouxDominique Bakry, Ivan Gentil & Michel Ledoux. 2014. Analysis and Geometry of Markov Diffusion Operators. Analysis and Geometry of Markov Diffusion Operators 235 275 .
Grigorios A. PavliotisGrigorios A. Pavliotis. 2014. Stochastic Processes and Applications. Stochastic Processes and Applications 87 137 .
Matthias Liero & Alexander Mielke. (2013) Gradient structures and geodesic convexity for reaction–diffusion systems. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 371:2005, pages 20120346.
Crossref
José Antonio Carrillo, Stéphane Cordier, Gustavo Deco & Simona Mancini. (2013) Complexity Reduction of Rate-Equations Models for Two-Choice Decision-Making. PLoS ONE 8:12, pages e80820.
Crossref
Chao Deng & Congming Li. (2013) Endpoint bilinear estimates and applications to the two-dimensional Poisson–Nernst–Planck system. Nonlinearity 26:11, pages 2993-3009.
Crossref
Jean Dolbeault & Giuseppe Toscani. (2013) Improved interpolation inequalities, relative entropy and fast diffusion equations. Annales de l'Institut Henri Poincaré C, Analyse non linéaire 30:5, pages 917-934.
Crossref
Julian Tugaut. (2013) Convergence to the equilibria for self-stabilizing processes in double-well landscape. The Annals of Probability 41:3A.
Crossref
Julian Tugaut. (2013) Self-stabilizing processes in multi-wells landscape in -convergence . Stochastic Processes and their Applications 123:5, pages 1780-1801.
Crossref
T. D. Frank. (2013) Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences. ISRN Mathematical Physics 2013, pages 1-28.
Crossref
Xinfu Chen, Carey Caginalp, Jianghao Hao & Yajing Zhang. (2013) Effects of white noise in multistable dynamics. Discrete & Continuous Dynamical Systems - B 18:7, pages 1805-1825.
Crossref
Xiuqing Chen & Jian-Guo Liu. (2013) Analysis of Polymeric Flow Models and Related Compactness Theorems in Weighted Spaces. SIAM Journal on Mathematical Analysis 45:3, pages 1179-1215.
Crossref
Jean Dolbeault, Maria J. Esteban, Michal Kowalczyk & Michael Loss. (2013) Sharp Interpolation Inequalities on the Sphere: New Methods and Consequences. Chinese Annals of Mathematics, Series B 34:1, pages 99-112.
Crossref
T. Leliévre. 2013. Numerical Mathematics and Advanced Applications 2011. Numerical Mathematics and Advanced Applications 2011 791 810 .
John W. Barrett & Endre Süli. (2012) Existence of global weak solutions to finitely extensible nonlinear bead–spring chain models for dilute polymers with variable density and viscosity. Journal of Differential Equations 253:12, pages 3610-3677.
Crossref
ANTON ARNOLD, IRENE M. GAMBA, MARIA PIA GUALDANI, STÉPHANE MISCHLER, CLEMENT MOUHOT & CHRISTOF SPARBER. (2012) THE WIGNER–FOKKER–PLANCK EQUATION: STATIONARY STATES AND LARGE TIME BEHAVIOR. Mathematical Models and Methods in Applied Sciences 22:11, pages 1250034.
Crossref
François Bolley, Ivan Gentil & Arnaud Guillin. (2012) Convergence to equilibrium in Wasserstein distance for Fokker–Planck equations. Journal of Functional Analysis 263:8, pages 2430-2457.
Crossref
Miku Hirata, Aya Nemoto & Hiroaki Yoshida. (2012) An Integral Representation of the Relative Entropy. Entropy 14:8, pages 1469-1477.
Crossref
JOHN W. BARRETT & ENDRE SÜLI. (2012) EXISTENCE AND EQUILIBRATION OF GLOBAL WEAK SOLUTIONS TO KINETIC MODELS FOR DILUTE POLYMERS II: HOOKEAN-TYPE MODELS. Mathematical Models and Methods in Applied Sciences 22:05, pages 1150024.
Crossref
José A. Cañizo, José A. Carrillo & Maria E. Schonbek. (2012) Decay rates for a class of diffusive-dominated interaction equations. Journal of Mathematical Analysis and Applications 389:1, pages 541-557.
Crossref
Bin Qian. (2012) A generalization of Hamilton’s gradient estimate. Journal of Geometry and Physics 62:5, pages 1064-1069.
Crossref
Claude Le Bris & Tony Lelièvre. (2012) Micro-macro models for viscoelastic fluids: modelling, mathematics and numerics. Science China Mathematics 55:2, pages 353-384.
Crossref
Lingbing He, Claude Le Bris & Tony Lelièvre. (2012) Periodic long-time behaviour for an approximate model of nematic polymers. Kinetic & Related Models 5:2, pages 357-382.
Crossref
Jean Dolbeault, Bruno Nazaret & Giuseppe Savaré. (2012) From Poincaré to Logarithmic Sobolev Inequalities: A Gradient Flow Approach. SIAM Journal on Mathematical Analysis 44:5, pages 3186-3216.
Crossref
Hailiang Liu & Hui Yu. (2012) An Entropy Satisfying Conservative Method for the Fokker–Planck Equation of the Finitely Extensible Nonlinear Elastic Dumbbell Model. SIAM Journal on Numerical Analysis 50:3, pages 1207-1239.
Crossref
Frédéric Legoll & Tony Lelièvre. 2012. Numerical Analysis of Multiscale Computations. Numerical Analysis of Multiscale Computations 279 329 .
José Antonio Carrillo, Stéphane Cordier & Simona Mancini. (2010) A decision-making Fokker–Planck model in computational neuroscience. Journal of Mathematical Biology 63:5, pages 801-830.
Crossref
T. Lelièvre & K. Minoukadeh. (2011) Long-Time Convergence of an Adaptive Biasing Force Method: The Bi-Channel Case. Archive for Rational Mechanics and Analysis 202:1, pages 1-34.
Crossref
Xiang-Dong Li. (2011) Perelmanʼs W-entropy for the Fokker–Planck equation over complete Riemannian manifolds. Bulletin des Sciences Mathématiques 135:6-7, pages 871-882.
Crossref
JOHN W. BARRETT & ENDRE SÜLI. (2011) EXISTENCE AND EQUILIBRATION OF GLOBAL WEAK SOLUTIONS TO KINETIC MODELS FOR DILUTE POLYMERS I: FINITELY EXTENSIBLE NONLINEAR BEAD-SPRING CHAINS. Mathematical Models and Methods in Applied Sciences 21:06, pages 1211-1289.
Crossref
M Ottobre & G A Pavliotis. (2011) Asymptotic analysis for the generalized Langevin equation. Nonlinearity 24:5, pages 1629-1653.
Crossref
Daniel Matthes, Ansgar Jüngel & Giuseppe Toscani. (2010) Convex Sobolev Inequalities Derived from Entropy Dissipation. Archive for Rational Mechanics and Analysis 199:2, pages 563-596.
Crossref
Christophe Chipot & Tony Lelièvre. (2011) Enhanced Sampling of Multidimensional Free-Energy Landscapes Using Adaptive Biasing Forces. SIAM Journal on Applied Mathematics 71:5, pages 1673-1695.
Crossref
John W. Barrett & Endre Süli. (2010) Finite element approximation of kinetic dilute polymer models with microscopic cut-off. ESAIM: Mathematical Modelling and Numerical Analysis 45:1, pages 39-89.
Crossref
Jean-Philippe Bartier, Adrien Blanchet, Jean Dolbeault & Miguel Escobedo. (2011) Improved intermediate asymptotics for the heat equation. Applied Mathematics Letters 24:1, pages 76-81.
Crossref
Renjun Duan, Massimo Fornasier & Giuseppe Toscani. (2010) A Kinetic Flocking Model with Diffusion. Communications in Mathematical Physics 300:1, pages 95-145.
Crossref
Chao Deng, Jihong Zhao & Shangbin Cui. (2010) Well-posedness of a dissipative nonlinear electrohydrodynamic system in modulation spaces. Nonlinear Analysis: Theory, Methods & Applications 73:7, pages 2088-2100.
Crossref
Manuel del Pino, Jean Dolbeault, Stathis Filippas & Achilles Tertikas. (2010) A logarithmic Hardy inequality. Journal of Functional Analysis 259:8, pages 2045-2072.
Crossref
Frédéric Legoll & Tony Lelièvre. (2010) Effective dynamics using conditional expectations. Nonlinearity 23:9, pages 2131-2163.
Crossref
J.-B. Maillet & G. Stoltz. (2010) Sampling Constraints in Average: The Example of Hugoniot Curves. Applied Mathematics Research eXpress.
Crossref
François Bolley & Ivan Gentil. (2010) Phi-entropy inequalities for diffusion semigroups. Journal de Mathématiques Pures et Appliquées 93:5, pages 449-473.
Crossref
Martin Burger, Marco Di Francesco, Jan-Frederik Pietschmann & Bärbel Schlake. (2010) Nonlinear Cross-Diffusion with Size Exclusion. SIAM Journal on Mathematical Analysis 42:6, pages 2842-2871.
Crossref
F. Salvarani. (2009) Asymptotic behavior for logarithmic diffusion. Journal of Mathematical Physics 50:11.
Crossref
Stefano Lisini. (2008) Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces. ESAIM: Control, Optimisation and Calculus of Variations 15:3, pages 712-740.
Crossref
Axel Klar, Philip Reuterswärd & Mohammed Seaïd. (2008) A Semi-Lagrangian Method for a Fokker-Planck Equation Describing Fiber Dynamics. Journal of Scientific Computing 38:3, pages 349-367.
Crossref
Thierry Gallay & Luis Miguel Rodrigues. (2009) Sur le temps de vie de la turbulence bidimensionnelle. Annales de la Faculté des sciences de Toulouse : Mathématiques 17:4, pages 719-733.
Crossref
Yann Ollivier. (2009) Ricci curvature of Markov chains on metric spaces. Journal of Functional Analysis 256:3, pages 810-864.
Crossref
Jean Dolbeault, Bruno Nazaret & Giuseppe Savaré. (2008) A new class of transport distances between measures. Calculus of Variations and Partial Differential Equations 34:2, pages 193-231.
Crossref
Adrien Blanchet, Jean Dolbeault & MichaŁ Kowalczyk. (2009) Stochastic Stokes' Drift, Homogenized Functional Inequalities, and Large Time Behavior of Brownian Ratchets. SIAM Journal on Mathematical Analysis 41:1, pages 46-76.
Crossref
Claude Le Bris & Tony Lelièvre. 2009. Multiscale Modeling and Simulation in Science. Multiscale Modeling and Simulation in Science 49 137 .
Till D. Frank. 2009. Encyclopedia of Complexity and Systems Science. Encyclopedia of Complexity and Systems Science 5239 5265 .
Marco Di Francesco, Klemens Fellner & Peter A Markowich. (2008) The entropy dissipation method for spatially inhomogeneous reaction–diffusion-type systems. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 464:2100, pages 3273-3300.
Crossref
Mihai Bostan & Thierry Goudon. (2008) High-electric-field limit for the Vlasov–Maxwell–Fokker–Planck system. Annales de l'Institut Henri Poincaré C, Analyse non linéaire 25:6, pages 1221-1251.
Crossref
Marco Di Francesco & Jesús Rosado. (2008) Fully parabolic Keller–Segel model for chemotaxis with prevention of overcrowding. Nonlinearity 21:11, pages 2715-2730.
Crossref
Christian Dogbé. (2008) On the entropy method for the hydrodynamical limit of some kinetic theory models. Mathematical and Computer Modelling 48:1-2, pages 179-190.
Crossref
A. Arnold, J.A. Carrillo & C. Klapproth. (2008) Improved entropy decay estimates for the heat equation. Journal of Mathematical Analysis and Applications 343:1, pages 190-206.
Crossref
JOHN W. BARRETT & ENDRE SÜLI. (2011) EXISTENCE OF GLOBAL WEAK SOLUTIONS TO DUMBBELL MODELS FOR DILUTE POLYMERS WITH MICROSCOPIC CUT-OFF. Mathematical Models and Methods in Applied Sciences 18:06, pages 935-971.
Crossref
Tony Lelièvre, Mathias Rousset & Gabriel Stoltz. (2008) Long-time convergence of an adaptive biasing force method. Nonlinearity 21:6, pages 1155-1181.
Crossref
Marco Di Francesco & Marcus Wunsch. (2008) Large time behavior in Wasserstein spaces and relative entropy for bipolar drift-diffusion-Poisson models. Monatshefte für Mathematik 154:1, pages 39-50.
Crossref
NICOLAS VAUCHELET. (2011) DIFFUSIVE TRANSPORT OF PARTIALLY QUANTIZED PARTICLES: L log L SOLUTIONS . Mathematical Models and Methods in Applied Sciences 18:04, pages 489-510.
Crossref
HAO WU, PETER A. MARKOWICH & SONGMU ZHENG. (2011) GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR FOR A SEMICONDUCTOR DRIFT-DIFFUSION-POISSON MODEL. Mathematical Models and Methods in Applied Sciences 18:03, pages 443-487.
Crossref
Martin Grothaus & Axel Klar. (2008) Ergodicity and Rate of Convergence for a Nonsectorial Fiber Lay-Down Process. SIAM Journal on Mathematical Analysis 40:3, pages 968-983.
Crossref
L. L. Bonilla, T. Götz, A. Klar, N. Marheineke & R. Wegener. (2008) Hydrodynamic Limit of a Fokker–Planck Equation Describing Fiber Lay-Down Processes. SIAM Journal on Applied Mathematics 68:3, pages 648-665.
Crossref
Ansgar Jüngel & Daniel Matthes. (2008) The Derrida–Lebowitz–Speer–Spohn Equation: Existence, NonUniqueness, and Decay Rates of the Solutions. SIAM Journal on Mathematical Analysis 39:6, pages 1996-2015.
Crossref
Grégoire Allaire, Anton Arnold, Pierre Degond & Thomas Yizhao HouAnton Arnold. 2008. Quantum Transport. Quantum Transport 45 109 .
Piotr Biler & Wojbor Woyczyński. (2007) General nonlocal diffusion–convection mean field models: Nonexistence of global solutions. Physica A: Statistical Mechanics and its Applications 379:2, pages 523-533.
Crossref
JEAN-PHILIPPE BARTIER, JEAN DOLBEAULT, REINHARD ILLNER & MICHAŁ KOWALCZYK. (2011) A QUALITATIVE STUDY OF LINEAR DRIFT-DIFFUSION EQUATIONS WITH TIME-DEPENDENT OR DEGENERATE COEFFICIENTS. Mathematical Models and Methods in Applied Sciences 17:03, pages 327-362.
Crossref
Luigi Ambrosio & Giuseppe Savaré. 2007. 1 136 .
Marco Di Francesco & Corrado Lattanzio. (2006) Optimal rate of decay to diffusion waves for the Hamer model of radiating gases . Applied Mathematics Letters 19:10, pages 1046-1052.
Crossref
Michael C. Mackey & Marta Tyran-Kamińska. (2006) Temporal Behavior of the Conditional and Gibbs’ Entropies. Journal of Statistical Physics 124:6, pages 1443-1470.
Crossref
Benjamin Jourdain, Claude Le Bris, Tony Lelièvre & Félix Otto. (2006) Long-Time Asymptotics of a Multiscale Model for Polymeric Fluid Flows. Archive for Rational Mechanics and Analysis 181:1, pages 97-148.
Crossref

Displaying 200 of 228 citing articles. Use the download link below to view the full list of citing articles.

Download full citations list

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.