Singular limit problem for the Keller–Segel system and drift–diffusion system in scaling critical spaces

2019 ◽  
Vol 20 (2) ◽  
pp. 421-457 ◽  
Author(s):  
Masaki Kurokiba ◽  
Takayoshi Ogawa
Keyword(s):  
Limit Problem ◽  
Singular Limit ◽  
Diffusion System ◽  
Drift Diffusion ◽  
Critical Spaces

scholarly journals Maximal regularity and a singular limit problem for the Patlak–Keller–Segel system in the scaling critical space involving BMO

2021 ◽  
Vol 3 (1) ◽  
Author(s):  
Masaki Kurokiba ◽  
Takayoshi Ogawa
Keyword(s):  
Function Space ◽  
Maximal Regularity ◽  
Limit Problem ◽  
Singular Limit ◽  
Time Parameter ◽  
Elliptic Type ◽  
Critical Space ◽  
Drift Diffusion ◽  
Critical Function ◽  
The Cauchy Problem

AbstractWe consider a singular limit problem of the Cauchy problem for the Patlak–Keller–Segel equation in a scaling critical function space. It is shown that a solution to the Patlak–Keller–Segel system in a scaling critical function space involving the class of bounded mean oscillations converges to a solution to the drift-diffusion system of parabolic-elliptic type (simplified Keller–Segel model) strongly as the relaxation time parameter $$\tau \rightarrow \infty $$ τ → ∞ . For the proof, we show generalized maximal regularity for the heat equation in the homogeneous Besov spaces and the class of bounded mean oscillations and we utilize them systematically as well as the continuous embeddings between the interpolation spaces $$\dot{B}^s_{q,\sigma }({\mathbb {R}}^n)$$ B ˙ q , σ s ( R n ) and $$\dot{F}^s_{q,\sigma }({\mathbb {R}}^n)$$ F ˙ q , σ s ( R n ) for the proof of the singular limit. In particular, end-point maximal regularity in BMO and space time modified class introduced by Koch–Tataru is utilized in our proof.

scholarly journals On a Drift–Diffusion System for Semiconductor Devices

2016 ◽  
Vol 17 (12) ◽  
pp. 3473-3498 ◽  
Author(s):  
Rafael Granero-Belinchón
Keyword(s):  
Semiconductor Devices ◽  
Diffusion System ◽  
Drift Diffusion

scholarly journals Uniform-in-time bounds for approximate solutions of the drift–diffusion system

2019 ◽  
Vol 141 (4) ◽  
pp. 881-916
Author(s):  
M. Bessemoulin-Chatard ◽  
C. Chainais-Hillairet
Keyword(s):  
Approximate Solutions ◽  
Diffusion System ◽  
Drift Diffusion ◽  
Time Bounds

Asymptotic Analysis of the Drift-Diffusion Equations and Hamilton–Jacobi Equations

1997 ◽  
Vol 07 (01) ◽  
pp. 61-80 ◽  
Author(s):  
Ph. Montarnal ◽  
B. Perthame
Keyword(s):  
Asymptotic Behavior ◽  
Coupled System ◽  
Limit Problem ◽  
Diffusion Equations ◽  
Drift Diffusion ◽  
Diffusion Term ◽  
Electronic Concentration ◽  
New Variables ◽  
Jacobi Equations ◽  
Original Approach

We study the asymptotic behavior of the semiconductor drift-diffusion (DD) equations with a vanishing diffusion term. In order to obtain a closed limit problem, we need to introduce two new variables involving the logarithm of the electronic concentration. We show that the limit problem is a coupled system of Hamilton–Jacobi equations and variational inequalities. In the mono-dimensional case, we show that this limit problem has a unique solution, which allows us to prove the convergence of the DD model for vanishing viscosities. Our method, which is an extension of previous asymptotic studies, sheds new light on the convergence of the electronic concentration, improves some necessary mathematical hypotheses and provides an original approach to the problem, well suited for numerical purposes.

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