2017/08/15 by Jukka Kemppainen, Kemppainen, Jukka, Rico Zacher +1
Engineering · Mathematics · Physics and Astronomy · #35R11 #45K05 #47G20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Fractional Differential Equations Solutions #Statistical Mechanics and Entropy
paper · pdf · doi:10.48550/arxiv.1708.04572
openalex publication_date 2017/08/15 · openalex created_date 2022/10/07 · openalex updated_date 2026/07/28
We consider a rather general class of non-local in time Fokker-Planck\nequations and show by means of the entropy method that as t\→ \∞ the\nsolution converges in L1 to the unique steady state. Important special cases\nare the time-fractional and ultraslow diffusion case. We also prove estimates\nfor the rate of decay. In contrast to the classical (local) case, where the\nusual time derivative appears in the Fokker-Planck equation, the obtained decay\nrate depends on the entropy, which is related to the integrability of the\ninitial datum. It seems that higher integrability of the initial datum leads to\nbetter decay rates and that the optimal decay rate is reached, as we show, when\nthe initial datum belongs to a certain weighted L2 space. We also show how\nour estimates can be adapted to the discrete-time case thereby improving known\ndecay rates from the literature.\n