A BDF2-approach for the non-linear Fokker-Planck equation

Simon Plazotta

doi:10.3934/dcds.2019120

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2019, Volume 39, Issue 5: 2893-2913. Doi: 10.3934/dcds.2019120

This issue Previous Article Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations Next Article A new proof of continuity of Lyapunov exponents for a class of

$ C^2 $

quasiperiodic Schrödinger cocycles without LDT

A BDF2-approach for the non-linear Fokker-Planck equation

Simon Plazotta

Boltzmannstra. 3, D-85747 Garching, Germany

Received: July 2018

Revised: October 2018

Published: January 2019

This research has been supported by the German Research Foundation (DFG), SFB TRR 109. The author would like to thank Daniel Matthes for helpful discussions and remarks

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We prove convergence of a variational formulation of the BDF2 method applied to the non-linear Fokker-Planck equation. Our approach is inspired by the JKO-method and exploits the differential structure of the underlying $ L^2 $-Wasserstein space. The technique presented here extends and strengthens the results of our own recent work [27] on the BDF2 method for general metric gradient flows in the special case of the non-linear Fokker-Planck equation: firstly, we do not require uniform semi-convexity of the augmented energy functional; secondly, we prove strong instead of merely weak convergence of the time-discrete approximations; thirdly, we directly prove without using the abstract theory of curves of maximal slope that the obtained limit curve is a weak solution of the non-linear Fokker-Planck equation.

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Mathematics Subject Classification: Primary: 34G25, 35A15, 35G25, 35K46, 65L06, 65J08.

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