scholarly journals Diffusion approximation for multi-scale stochastic reaction-diffusion equations

2021 ◽  
Vol 300 ◽  
pp. 155-184
Author(s):  
Longjie Xie ◽  
Li Yang
Keyword(s):  
Diffusion Approximation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Multi Scale ◽  
Stochastic Reaction

A BGK MODEL FOR CHEMICAL PROCESSES: THE REACTION-DIFFUSION APPROXIMATION

1994 ◽  
Vol 04 (01) ◽  
pp. 35-47 ◽  
Author(s):  
RENATO SPIGLER ◽  
DAMIÁN H. ZANETTE
Keyword(s):  
Kinetic Model ◽  
Diffusion Approximation ◽  
Reaction Diffusion ◽  
Chemical Processes ◽  
Reaction Diffusion Equations ◽  
The Other ◽  
Diffusion Equations ◽  
Bgk Model ◽  
Chemical Substances ◽  
Chemical Effects

A BGK-type kinetic model is derived for describing the interaction of chemical substances. The ensuing equation is then solved asymptotically on certain space-time scales on which an appreciable interplay between kinetic and chemical effects, or the prevailing of one on the other, can be observed. The description of the interaction at the macroscopic level consists of a hierarchy of reaction-diffusion equations satisfied by the densities. Comparison is made with similar results previously obtained from certain phenomenological models, and illustrative examples are given.

scholarly journals Averaging Principles for Nonautonomous Two-Time-Scale Stochastic Reaction-Diffusion Equations with Jump

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-22
Author(s):  
Yong Xu ◽  
Ruifang Wang
Keyword(s):  
Reaction Diffusion ◽  
Evolution Family ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Averaging Principle ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Random Measures ◽  
Fast System ◽  
Stochastic Reaction

In this paper, we aim to develop the averaging principle for a slow-fast system of stochastic reaction-diffusion equations driven by Poisson random measures. The coefficients of the equation are assumed to be functions of time, and some of them are periodic or almost periodic. Therefore, the Poisson term needs to be processed, and a new averaged equation needs to be given. For this reason, the existence of time-dependent evolution family of measures associated with the fast equation is studied and proved that it is almost periodic. Next, according to the characteristics of almost periodic functions, the averaged coefficient is defined by the evolution family of measures, and the averaged equation is given. Finally, the validity of the averaging principle is verified by using the Khasminskii method.

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