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Conservative-dissipative approximation schemes for a generalized Kramers equation. (English) Zbl 1370.35165
Summary: We propose three new discrete variational schemes that capture the conservative-dissipative structure of a generalized Kramers equation. The first two schemes are single-step minimization schemes, whereas the third one combines a streaming and a minimization step. The cost functionals in the schemes are inspired by the rate functional in the Freidlin-Wentzell theory of large deviations for the underlying stochastic system. We prove that all three schemes converge to the solution of the generalized Kramers equation.

35K15 Initial value problems for second-order parabolic equations
35A15 Variational methods applied to PDEs
60J60 Diffusion processes
65C30 Numerical solutions to stochastic differential and integral equations
Full Text: DOI arXiv
[1] Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica 7 pp 284– (1940) · Zbl 0061.46405
[2] Jordan, Free energy and the Fokker-Planck equation, Physica D. Nonlinear Phenomena 107 (2-4) pp 265– (1997) · Zbl 1029.82507
[3] Jordan, The variational formulation of the fokker-planck equation, SIAM Journal on Mathematical Analysis 29 (1) pp 1– (1998) · Zbl 0915.35120
[4] Mielke, Handb. Differ. Equ II, in: Evolutionary equations pp 461– (2005)
[5] Mielke, A variational formulation of rate-independent phase transformations using an extremum principle, Archive for Rational Mechanics and Analysis 162 (2) pp 137– (2002) · Zbl 1012.74054
[6] Ambrosio, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2. ed. (2008) · Zbl 1210.28005
[7] Carrillo, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Revista Matemática Iberoamericana 19 (3) pp 971– (2003) · Zbl 1073.35127
[8] Düring, A gradient flow scheme for nonlinear fourth order equations, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences 14 (3) pp 935– (2010)
[9] Arnrich, Passing to the limit in a Wasserstein gradient flow: from diffusion to reaction, Calculus of Variations and Partial Differential Equations 44 pp 419– (2012) · Zbl 1270.35055
[10] Sandier, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Communications on Pure and Applied Mathematics 57 (12) pp 1627– (2004) · Zbl 1065.49011
[11] Stefanelli, The Brezis-Ekeland principle for doubly nonlinear equations, SIAM Journal on Control and Optimization 47 pp 1615– (2008) · Zbl 1194.35214
[12] Huang, A variational principle for the Kramers equation with unbounded external forces, Journal of Mathematical Analysis and Applications 250 (1) pp 333– (2000) · Zbl 0971.82031
[13] Huang C A variational principle for a class of ultraparabolic equations 2011
[14] Adams, From a large-deviations principle to the Wasserstein gradient flow: a new micro-macro passage, Communications in Mathematical Physics 307 pp 791– (2011) · Zbl 1267.60106
[15] Dirr, Upscaling from particle models to entropic gradient flows, Journal of Mathematics and Physics 53 (6) pp 063– (2012) · Zbl 1280.35009
[16] Duong, Wasserstein gradient flows from large deviations of many-particle limits, ESAIM: Control, Optimisation and Calculus of Variations, E-First (2013) · Zbl 1284.35011
[17] Duong MH Peletier MA Zimmer J GENERIC formalism of a Vlasov-Fokker-Planck equation and connection to large-deviation principles (submitted) 2013 http://arxiv.org/abs/1302.1024
[18] Peletier, Variational formulation of the Fokker-Planck equation with decay: a particle approach, Communications in Contemporary Mathematics · Zbl 1280.35156
[19] Dembo, Large Deviations Techniques and Applications, volume 38 of Stochastic Modelling and Applied Probability, 2. ed. (1987) · Zbl 0793.60030
[20] Villani, Topics in optimal transportation 58 of (2003) · Zbl 1106.90001
[21] Öttinger, Beyond Equilibrium Thermodynamics, 1. ed. (2005)
[22] Holden, Splitting Methods for Partial Differential Equations with Rough Solutions: Analysis and MATLAB Programs (2010) · Zbl 1191.35005
[23] Delarue, Density estimates for a random noise propagating through a chain of differential equations, Journal of Functional Analysis 259 (6) pp 1577– (2010) · Zbl 1223.60037
[24] Gangbo, Optimal transport for the system of isentropic Euler equations, Communications in Partial Differential Equations 34 (7-9) pp 1041– (2009) · Zbl 1182.35161
[25] Westdickenberg, Projections onto the cone of optimal transport maps and compressible fluid flows, Journal of Hyperbolic Differential Equations 7 (4) pp 605– (2010) · Zbl 1213.35304
[26] Chavanis, Generalized thermodynamics and fokker-planck equations: applications to stellar dynamics and two-dimensional turbulence, Physical Review E 68 pp 036– (2003)
[27] Chavanis, Chapman-Enskog derivation of the generalized smoluchowski equation, Physica A: Statistical Mechanics and its Applications 341 pp 145– (2004)
[28] Chavanis, Statistical mechanics of two dimensional vortices and collisionless stellar systems, The Astrophysical Journal 471 pp 385– (1996)
[29] Renesse, On optimal transport view on Schrödinger’s equation, Canadian Mathematical Bulletin 55 (4) pp 858– (2012) · Zbl 1256.81072
[30] Ambrosio, Hamiltonian ODEs in the Wasserstein space of probability measures, Communications on Pure and Applied Mathematics 61 (1) pp 18– (2008) · Zbl 1132.37028
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