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The Poisson-Nernst-Planck system of equations used to model ionic transport is interpreted as a gradient flow for the Wasserstein distance and a free energy in the space of probability measures with finite second moment.
Singular integrals and differentiability properties of functions
Elias M. Stein · 1970
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Compact sets in the space L p ( 0 , T , B ) L^{p}(0,T;B)
Jacques Simon · 1987
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Semiconductor equations
P. A. Markowich, C. A. Ringhofer, and C. Schmeiser · 1990
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On the uniqueness of solutions to the drift-diffusion model of semiconductor devices
Herbert Gajewski · 1994
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Harmonic analysis techniques for second order elliptic boundary value problems
Carlos E. Kenig · 1994
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On the time-dependent drift-diffusion model for semiconductors
Weifu Fang and Kazufumi Ito · 1995
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A nonlinear drift-diffusion system with electric convection arising in electrophoretic and semiconductor modeling
Ansgar Jüngel · 1997
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A convexity principle for interacting gases
Robert J. McCann · 1997
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The variational formulation of the Fokker-Planck equation
Richard Jordan, David Kinderlehrer, and Felix Otto · 1998
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Dynamics of labyrinthine pattern formation in magnetic fluids: a mean-field theory
Felix Otto · 1998
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Large time asymptotics of nonlinear drift-diffusion systems with Poisson coupling
Piotr Biler, Jean Dolbeault, and Peter A. Markowich · 1999
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On large time asymptotics for drift-diffusion-Poisson systems
Anton Arnold, Peter Markowich, and Giuseppe Toscani · 2000
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A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem
Jean-David Benamou and Yann Brenier · 2000
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Long time behavior of solutions of Nernst-Planck and Debye-Hückel drift-diffusion systems
Piotr Biler and Jean Dolbeault · 2000
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On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations
Anton Arnold, Peter Markowich, Giuseppe Toscani, and Andreas Unterreiter · 2001
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Quasi-hydrodynamic semiconductor equations
Ansgar Jüngel · 2001
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Analysis
Elliott H. Lieb and Michael Loss · 2001
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The geometry of dissipative evolution equations: the porous medium equation
Felix Otto · 2001
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Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates
José A. Carrillo, Robert J. McCann, and Cédric Villani · 2003
Well-posedness for the drift-diffusion system in L p L^{p} arising from the semiconductor device simulation
Masaki Kurokiba and Takayoshi Ogawa · 2008
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The Janossy effect and hybrid variational principles
David Kinderlehrer and Michal Kowalczyk · 2009
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A family of nonlinear fourth order equations of gradient flow type
Daniel Matthes, Robert J. McCann, and Giuseppe Savaré · 2009
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A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems
Alexander Mielke · 2011
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A user’s guide to optimal transport
Luigi Ambrosio and Nicola Gigli · 2013
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The parabolic-parabolic Keller-Segel system with critical diffusion as a gradient flow in ℝ d , d ≥ 3 \mathbb{R}^{d},\ d\geq 3
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Topics in optimal transportation
Cédric Villani · 2003
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Diffusion mediated transport in multiple state systems
Stuart Hastings, David Kinderlehrer, and J. Bryce McLeod · 2007
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The porous medium equation
Juan Luis Vázquez · 2007
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Gradient flows in metric spaces and in the space of probability measures
Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré · 2008
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Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model
Adrien Blanchet, Vincent Calvez, and José A. Carrillo · 2008
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Large time behavior in Wasserstein spaces and relative entropy for bipolar drift-diffusion-Poisson models
Marco Di Francesco and Marcus Wunsch · 2008
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Adrien Blanchet and Philippe Laurençot · 2013
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A gradient flow approach to a thin film approximation of the Muskat problem
Philippe Laurençot and Bogdan-Vasile Matioc · 2013
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New porous medium Poisson-Nernst-Planck equations for strongly oscillating electric potentials
M. Schmuck · 2013
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A hybrid variational principle for the keller-segel system in ℝ 2 \mathbb{R}^{2}
Adrien Blanchet, José Antonio Carrillo, David Kinderlehrer, Michal Kowalczyk, Philippe Laurençot, and Stefano Lisini · 2014
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Exponential convergence to equilibrium in a coupled gradient flow system modelling chemotaxis
Daniel Matthes and Jonathan Zinsl · 2014
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An energetic variational approach for ion transport
Shixin Xu, Ping Sheng, and Chun Liu · 2014
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Existence of solutions for a nonlinear system of parabolic equations with gradient flow structure
Jonathan Zinsl · 2014
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