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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.

MSC:
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
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References:
[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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