scholarly journals Classical solutions to the Fokker–Planck–BGK equation with infinite energy

2016 ◽  
Vol 437 (2) ◽  
pp. 816-840
Author(s):  
Zili Chen
Keyword(s):  
Classical Solutions ◽  
Bgk Equation ◽  
Infinite Energy ◽  
Fokker Planck

THE GLOBAL CLASSICAL SOLUTION OF THE VLASOV–MAXWELL–FOKKER–PLANCK SYSTEM NEAR MAXWELLIAN

2011 ◽  
Vol 21 (05) ◽  
pp. 1007-1025 ◽  
Author(s):  
MYEONGJU CHAE
Keyword(s):  
Cauchy Problem ◽  
Electromagnetic Field ◽  
Classical Solution ◽  
Charged Particles ◽  
Classical Solutions ◽  
Global Classical Solution ◽  
Self Consistent ◽  
The Cauchy Problem ◽  
Fokker Planck

The Vlasov–Maxwell–Fokker–Planck system is used in modeling distribution of charged particles in plasma, where particles interact via collisions and through their self-consistent electromagnetic field. We prove the existence of global in time classical solutions to the Cauchy problem near Maxwellians.

scholarly journals The model reduction of the Vlasov-Poisson-Fokker-Planck system to the Poisson-Nernst-Planck system via the Deep Neural Network Approach

2021 ◽  
Author(s):  
Jae Yong Lee ◽  
Jin Woo Jang ◽  
Hyung Ju Hwang
Keyword(s):  
Neural Network ◽  
Boundary Condition ◽  
Model Reduction ◽  
Deep Neural Network ◽  
Specular Reflection ◽  
Classical Solutions ◽  
Diffusion Limit ◽  
Flux Boundary Condition ◽  
Bounded Interval ◽  
Fokker Planck

The model reduction of a mesoscopic kinetic dynamics to a macroscopic continuum dynamics has been one of the fundamental questions in mathematical physics since Hilbert's time. In this paper, we consider a diagram of the diffusion limit from the Vlasov-Poisson-Fokker-Planck (VPFP) system on a bounded interval with the specular reflection boundary condition to the Poisson-Nernst–Planck (PNP) system with the no-flux boundary condition. We provide a Deep Learning algorithm to simulate the VPFP system and the PNP system by computing the time-asymptotic behaviors of the solution and the physical quantities. We analyze the convergence of the neural network solution of the VPFP system to that of the PNP system via the Asymptotic-Preserving (AP) scheme. Also, we provide several theoretical evidence that the Deep Neural Network (DNN) solutions to the VPFP and the PNP systems converge to the a priori classical solutions of each system if the total loss function vanishes.

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