Global existence for a nonlocal and nonlinear Fokker–Planck equation

• Published in: Zeitschrift für angewandte Mathematik und Physik Vol. 66; no. 2; pp. 293 - 315
• Main Authors: Dreyer, Wolfgang, Huth, Robert, Mielke, Alexander, Rehberg, Joachim, Winkler, Michael
• Format: Journal Article
• Language: English
• Published: Basel Springer Basel 01.04.2015
• Subjects: Constants; Engineering; Fokker-Planck equation; Functions (mathematics); Intervals; Lagrange multipliers; Mathematical analysis; Mathematical Methods in Physics; Nonlinearity; Steady state; Theoretical and Applied Mechanics; Gradient flow with time-dependent constraint; 49Jxx; Energy-dissipation relation; Fokker–Planck equation; 76Dxx; 76Txx; 35C20; 35B40
• ISSN: 0044-2275; 1420-9039
• DOI: 10.1007/s00033-014-0401-1

We consider a Fokker–Planck equation on a compact interval where, as a constraint, the first moment is a prescribed function of time. Eliminating the associated Lagrange multiplier, one obtains nonlinear and nonlocal terms. After establishing suitable local existence results, we use the relative entropy as an energy functional. However, the time-dependent constraint leads to a source term such that a delicate analysis is needed to show that the dissipation terms are strong enough to control the work done by the constraint. We obtain global existence of solutions as long as the prescribed first moment stays in the interior of an interval. If the prescribed moment converges to a constant value inside the interior of the interval, then the solution stabilises to the unique steady state.