Abstract
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.
Similar content being viewed by others
References
Biler P., Hebisch W., Nadzieja T.: The debye system: existence and large time behavior of solution. Nonlinear Anal. TMA 23(9), 1189–1209 (1994)
Dreyer, W., Guhlke, C.: Herrmann, M.: Hysteresis and phase transition in many-particle storage systems. Continuum Mech. Thermodyn., pp. 1–21 (2011)
Dreyer W., Jamnik J., Guhlke C., Huth R., Moskon J., Gaberscek M.: The thermodynamic origin of hysteresis in insertion batteries. Nat. Mater. 9(5), 448–453 (2010)
Evans, L.C.: Partial Differential Equations, Volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (1998)
Glitzky A., Hünlich R.: Global estimates and asymptotics for electro-reaction-diffusion systems in heterostructures Appl. Anal. 66(3–4) 205–226 (1997)
Glitzky A., Merz W.: Single dopant diffusion in semiconductor technology. Math. Methods Appl. Sci. (MMAS) 27(2), 133–154 (2004)
Haskovec, J., Markowich, P.A., Mielke, A.: On uniform decay of the entropy for reaction-diffusion systems. J. Dyn. Diff. Equ. Submitted. WIAS Preprint 1768, (2013)
Herrmann, M. Niethammer, B. Velázquez, J.J.L.: Kramers and non-Kramers phase transitions in many-particle systems with dynamical constraint. SIAM Multiscale Model. Simul. 10(3), 818–852 (2012)
Herrmann, M., Niethammer, B., Velázquez, J.J.L.: Rate-independent dynamics and Kramers-type phase transitions in nonlocal Fokker-Planck equations with dynamical control. ArXiv e-prints, 2012. arXiv:1212.3128
Jordan R., Kinderlehrer D., Otto F.: The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998)
Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems, Volume 36 of Progress in Nonlinear Differential Equations and their Applications. Birkhäuser (1995)
Mielke A.: A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems. Nonlinearity 24(4), 1329 (2011)
Mielke A., Truskinovsky L.: From discrete visco-elasticity to continuum rate-independent plasticity: rigorous results. Arch. Ration. Mech. Anal. 203(2), 577–619 (2012)
Otto F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26, 101–174 (2001)
Visintin A.: Strong convergence results related to strict convexity. Commun. Partial Differ. Equ. 9(5), 439–466 (1984)
Zheng, S.: Nonlinear Evolution Equations. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics. Chapman & Hall/CRC Press (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dreyer, W., Huth, R., Mielke, A. et al. Global existence for a nonlocal and nonlinear Fokker–Planck equation. Z. Angew. Math. Phys. 66, 293–315 (2015). https://doi.org/10.1007/s00033-014-0401-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00033-014-0401-1