Singular Limit Problem to the Keller-Segel System in Critical Spaces and Related Medical Problems—An Application of Maximal Regularity

2021 ◽  
pp. 103-182
Author(s):  
Takayoshi Ogawa
Keyword(s):  
Maximal Regularity ◽  
Limit Problem ◽  
Singular Limit ◽  
Critical Spaces ◽  
Medical Problems

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.

Scale analysis of a hydrodynamic model of plasma

2014 ◽  
Vol 25 (02) ◽  
pp. 371-394 ◽  
Author(s):  
Donatella Donatelli ◽  
Eduard Feireisl ◽  
Antonín Novotný
Keyword(s):  
Hydrodynamic Model ◽  
Debye Length ◽  
Careful Analysis ◽  
Euler System ◽  
Limit Problem ◽  
Singular Limit ◽  
Scale Analysis ◽  
Small Ratio ◽  
High Reynolds ◽  
Incompressible Euler

We examine a hydrodynamic model of the motion of ions in plasma in the regime of small Debye length, a small ratio of the ion/electron temperature, and high Reynolds number. We analyze the associated singular limit and identify the limit problem — the incompressible Euler system. The result leans on careful analysis of the oscillatory component of the solutions by means of Fourier analysis.

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