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  1. Calculus of Variations and Partial Differential Equations
  2. Calculus of Variations and Partial Differential Equations : Volume 44
  3. Calculus of Variations and Partial Differential Equations : Volume 44, Issue 3-4, July 2012
  4. Passing to the limit in a Wasserstein gradient flow: from diffusion to reaction
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Calculus of Variations and Partial Differential Equations : Volume 56
Calculus of Variations and Partial Differential Equations : Volume 55
Calculus of Variations and Partial Differential Equations : Volume 54
Calculus of Variations and Partial Differential Equations : Volume 53
Calculus of Variations and Partial Differential Equations : Volume 52
Calculus of Variations and Partial Differential Equations : Volume 51
Calculus of Variations and Partial Differential Equations : Volume 50
Calculus of Variations and Partial Differential Equations : Volume 49
Calculus of Variations and Partial Differential Equations : Volume 48
Calculus of Variations and Partial Differential Equations : Volume 47
Calculus of Variations and Partial Differential Equations : Volume 46
Calculus of Variations and Partial Differential Equations : Volume 45
Calculus of Variations and Partial Differential Equations : Volume 44
Calculus of Variations and Partial Differential Equations : Volume 44, Issue 3-4, July 2012
Volume-constrained minimizers for the prescribed curvature problem in periodic media
Weak KAM Theory topics in the stationary ergodic setting
Heat kernels of two-dimensional magnetic Schrödinger and Pauli operators
Entropy method for line-energies
Passing to the limit in a Wasserstein gradient flow: from diffusion to reaction
On the characterization of the compact embedding of Sobolev spaces
Local Poincaré inequalities from stable curvature conditions on metric spaces
Quasistatic evolution for Cam-Clay plasticity: properties of the viscosity solution
Beyond the Trudinger-Moser supremum
The geometric Neumann problem for the Liouville equation
Partial regularity of stable solutions to the Emden equation
Calculus of Variations and Partial Differential Equations : Volume 44, Issue 1-2, May 2012
Calculus of Variations and Partial Differential Equations : Volume 43
Calculus of Variations and Partial Differential Equations : Volume 42
Calculus of Variations and Partial Differential Equations : Volume 41
Calculus of Variations and Partial Differential Equations : Volume 40
Calculus of Variations and Partial Differential Equations : Volume 39
Calculus of Variations and Partial Differential Equations : Volume 38
Calculus of Variations and Partial Differential Equations : Volume 37
Calculus of Variations and Partial Differential Equations : Volume 36
Calculus of Variations and Partial Differential Equations : Volume 35
Calculus of Variations and Partial Differential Equations : Volume 34
Calculus of Variations and Partial Differential Equations : Volume 33
Calculus of Variations and Partial Differential Equations : Volume 32
Calculus of Variations and Partial Differential Equations : Volume 31
Calculus of Variations and Partial Differential Equations : Volume 30
Calculus of Variations and Partial Differential Equations : Volume 29
Calculus of Variations and Partial Differential Equations : Volume 28
Calculus of Variations and Partial Differential Equations : Volume 27
Calculus of Variations and Partial Differential Equations : Volume 26
Calculus of Variations and Partial Differential Equations : Volume 25
Calculus of Variations and Partial Differential Equations : Volume 24
Calculus of Variations and Partial Differential Equations : Volume 23
Calculus of Variations and Partial Differential Equations : Volume 22
Calculus of Variations and Partial Differential Equations : Volume 21
Calculus of Variations and Partial Differential Equations : Volume 20
Calculus of Variations and Partial Differential Equations : Volume 19
Calculus of Variations and Partial Differential Equations : Volume 18
Calculus of Variations and Partial Differential Equations : Volume 17
Calculus of Variations and Partial Differential Equations : Volume 16
Calculus of Variations and Partial Differential Equations : Volume 15
Calculus of Variations and Partial Differential Equations : Volume 14
Calculus of Variations and Partial Differential Equations : Volume 13
Calculus of Variations and Partial Differential Equations : Volume 12
Calculus of Variations and Partial Differential Equations : Volume 11
Calculus of Variations and Partial Differential Equations : Volume 10
Calculus of Variations and Partial Differential Equations : Volume 9
Calculus of Variations and Partial Differential Equations : Volume 8
Calculus of Variations and Partial Differential Equations : Volume 7
Calculus of Variations and Partial Differential Equations : Volume 6
Calculus of Variations and Partial Differential Equations : Volume 5

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Passing to the limit in a Wasserstein gradient flow: from diffusion to reaction

Content Provider SpringerLink
Author Arnrich, Steffen Peletier, Mark A. Mielke, Alexander Veneroni, Marco Savaré, Giuseppe
Copyright Year 2011
Abstract We study a singular-limit problem arising in the modelling of chemical reactions. At finite $${\varepsilon\, > \,0,}$$ the system is described by a Fokker-Planck convection-diffusion equation with a double-well convection potential. This potential is scaled by $${1 / \varepsilon,}$$ and in the limit $${\varepsilon\to0,}$$ the solution concentrates onto the two wells, resulting into a limiting system that is a pair of ordinary differential equations for the density at the two wells. This convergence has been proved in Peletier et al. (SIAM J Math Anal, 42(4):1805–1825, 2010), using the linear structure of the equation. In this study we re-prove the result by using solely the Wasserstein gradient-flow structure of the system. In particular we make no use of the linearity, nor of the fact that it is a second-order system. The first key step in this approach is a reformulation of the equation as the minimization of an action functional that captures the property of being a curve of maximal slope in an integrated form. The second important step is a rescaling of space. Using only the Wasserstein gradient-flow structure, we prove that the sequence of rescaled solutions is pre-compact in an appropriate topology. We then prove a Gamma-convergence result for the functional in this topology, and we identify the limiting functional and the differential equation that it represents. A consequence of these results is that solutions of the $${\varepsilon}$$ -problem converge to a solution of the limiting problem.
Ending Page 454
Page Count 36
Starting Page 419
File Format PDF
ISSN 09442669
e-ISSN 14320835
Journal Calculus of Variations and Partial Differential Equations
Issue Number 3-4
Volume Number 44
Language English
Publisher Springer-Verlag
Publisher Date 2011-08-30
Publisher Place Berlin, Heidelberg
Access Restriction One Nation One Subscription (ONOS)
Subject Keyword Homogenization; equations in media with periodic structure Initial-boundary value problems for second-order parabolic equations Variational methods Systems Theory, Control Theoretical, Mathematical and Computational Physics Second-order parabolic equations Problems with friction Singular perturbations Large deviations Singular parabolic equations Calculus of Variations and Optimal Control; Optimization Analysis General theory, nonlinear semigroups, evolution equations Reaction-diffusion equations
Content Type Text
Resource Type Article
Subject Applied Mathematics Analysis
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