Global Existence and Decay Rates of the Solutions Near Maxwellian for Non-linear Fokker–Planck Equations

2018 ◽  
Vol 173 (1) ◽  
pp. 222-241 ◽  
Author(s):  
Jie Liao ◽  
Qianrong Wang ◽  
Xiongfeng Yang
Keyword(s):  
Global Existence ◽  
Decay Rates ◽  
Non Linear ◽  
Fokker Planck Equations ◽  
Fokker Planck

scholarly journals Non-Linear Langevin and Fractional Fokker–Planck Equations for Anomalous Diffusion by Lévy Stable Processes

Entropy ◽  
2018 ◽  
Vol 20 (10) ◽  
pp. 760 ◽  
Author(s):  
Johan Anderson ◽  
Sara Moradi ◽  
Tariq Rafiq
Keyword(s):  
Anomalous Diffusion ◽  
Transport Coefficient ◽  
Numerical Solutions ◽  
Distribution Functions ◽  
Space Distribution ◽  
Non Linear ◽  
Non Local ◽  
Local Transport ◽  
Fokker Planck Equations ◽  
Fokker Planck

The numerical solutions to a non-linear Fractional Fokker–Planck (FFP) equation are studied estimating the generalized diffusion coefficients. The aim is to model anomalous diffusion using an FFP description with fractional velocity derivatives and Langevin dynamics where Lévy fluctuations are introduced to model the effect of non-local transport due to fractional diffusion in velocity space. Distribution functions are found using numerical means for varying degrees of fractionality of the stable Lévy distribution as solutions to the FFP equation. The statistical properties of the distribution functions are assessed by a generalized normalized expectation measure and entropy and modified transport coefficient. The transport coefficient significantly increases with decreasing fractality which is corroborated by analysis of experimental data.

scholarly journals Long-time behavior of non-local in time Fokker–Planck equations via the entropy method

2019 ◽  
Vol 29 (02) ◽  
pp. 209-235 ◽  
Author(s):  
Jukka Kemppainen ◽  
Rico Zacher
Keyword(s):  
Decay Rate ◽  
Entropy Method ◽  
Decay Rates ◽  
Time Behavior ◽  
Optimal Decay ◽  
Local Case ◽  
Special Cases ◽  
Non Local ◽  
Fokker Planck Equations ◽  
Fokker Planck

We consider a rather general class of non-local in time Fokker–Planck equations and show by means of the entropy method that as [Formula: see text], the solution converges in [Formula: see text] to the unique steady state. Important special cases are the time-fractional and ultraslow diffusion case. We also prove estimates for the rate of decay. In contrast to the classical (local) case, where the usual time derivative appears in the Fokker–Planck equation, the obtained decay rate depends on the entropy, which is related to the integrability of the initial datum. It seems that higher integrability of the initial datum leads to better decay rates and that the optimal decay rate is reached, as we show, when the initial datum belongs to a certain weighted [Formula: see text] space. We also show how our estimates can be adapted to the discrete-time case thereby improving known decay rates from the literature.

scholarly journals φ-Entropies: convexity, coercivity and hypocoercivity for Fokker–Planck and kinetic Fokker–Planck equations

2018 ◽  
Vol 28 (13) ◽  
pp. 2637-2666 ◽  
Author(s):  
Jean Dolbeault ◽  
Xingyu Li
Keyword(s):  
Planck Equation ◽  
Decay Rates ◽  
Point Of View ◽  
Diffusion Equations ◽  
Estimates Of Solutions ◽  
Optimal Decay Rates ◽  
Optimal Decay ◽  
Fokker Planck Equations ◽  
Kinetic Level ◽  
Fokker Planck

This paper is devoted to [Formula: see text]-entropies applied to Fokker–Planck and kinetic Fokker–Planck equations in the whole space, with confinement. The so-called [Formula: see text]-entropies are Lyapunov functionals which typically interpolate between Gibbs entropies and [Formula: see text] estimates. We review some of their properties in the case of diffusion equations of Fokker–Planck type, give new and simplified proofs, and then adapt these methods to a kinetic Fokker–Planck equation acting on a phase space with positions and velocities. At kinetic level, since the diffusion only acts on the velocity variable, the transport operator plays an essential role in the relaxation process. Here we adopt the [Formula: see text] point of view and establish a sharp decay rate. Rather than giving general but quantitatively vague estimates, our goal here is to consider simple cases, benchmark available methods and obtain sharp estimates on a key example. Some [Formula: see text]-entropies give rise to improved entropy–entropy production inequalities and, as a consequence, to faster decay rates for entropy estimates of solutions to non-degenerate diffusion equations. We prove that faster entropy decay also holds at kinetic level away from equilibrium and that optimal decay rates are achieved only in asymptotic regimes.

Stability of Global Maxwellian For Non-Linear Vlasov-Poisson-Fokker-Planck Equations

2019 ◽  
Vol 39 (1) ◽  
pp. 127-138
Author(s):  
Jie Liao ◽  
Qianrong Wang ◽  
Xiongfeng Yang
Keyword(s):  
Non Linear ◽  
Fokker Planck Equations ◽  
Fokker Planck

scholarly journals Gevrey hypoellipticity for linear and non-linear Fokker–Planck equations

2009 ◽  
Vol 246 (1) ◽  
pp. 320-339 ◽  
Author(s):  
Hua Chen ◽  
Wei-Xi Li ◽  
Chao-Jiang Xu
Keyword(s):  
Non Linear ◽  
Fokker Planck Equations ◽  
Fokker Planck
Sign in / Sign up

Export Citation Format

Share Document