Maximal regularity and a singular limit problem for the Patlak-Keller-Segel system in the scaling critical space involving \textit{BMO}
type: Item
instance of: scholarly article
title: Maximal regularity and a singular limit problem for the Patlak-Keller-Segel system in the scaling critical space involving \textit{BMO}
zbMATH Open document ID: 1487.35037
author: Masaki Kurokiba
author: Takayoshi Ogawa
published in: SN Partial Differential Equations and Applications
publication date: 2022-01-03T00:00:00Z
review text: The authors study the singular limit of the Cauchy problem for the parabolic-parabolic Patlak-Keller-Segel system in \(\mathbb{R}^n\), \(n \geq 2\), \[ \begin{cases} \partial_tu_{\tau} - \Delta u_{\tau} + \nabla \cdot (u_{\tau}\nabla \psi_{\tau}) = 0, & t>0, \ x \in \mathbb{R}^n,\\ \frac{1}{\tau} \partial_t \psi_{\tau} - \Delta \psi_{\tau} + \lambda \psi_{\tau} = u_{\tau}, & t>0, \ x \in \mathbb{R}^n,\\ u_{\tau}(0,x) = u_0, \psi_{\tau}(0,x) = \psi_0(x), & t=0, \ x \in \mathbb{R}^n, \end{cases}\tag{1} \] as the relaxation time \(\tau \rightarrow \infty\). Here in the above equation \(u_{\tau}, \psi_{\tau}\) denotes the density of molds and the concentration of the chemical substances respectively and \(\lambda \geq 0\). Formally, the singular limit \(u = \lim_{\tau \rightarrow \infty} u_{\tau}, \psi = \lim_{\tau \rightarrow \infty} \psi_{\tau}\), if exists, would expected to satisfy the parabolic-elliptic Patlak-Keller-Segel system \[ \begin{cases} \partial_t u - \Delta u + \nabla \cdot (u \nabla \psi) = 0, & t >0, \ x \in \mathbb{R}^n \\ -\Delta \psi + \lambda \psi = u, & t>0, \ x \in \mathbb{R}^n \\ u(0,x) = u_0(x), & t =0, \ x \in \mathbb{R}^n. \end{cases}\tag{2} \] One of the main obstruction in the singular limit problem is, in general, the solution space for \(\psi_{\tau}\) and \(\psi\) are different from each other. The authors rigorously justified the above formal claim in the scaling invariant critical spaces of Bochner-Lebesgue type: \begin{align*} (u_0, \psi_0) \in \Big(L^{\frac{n}{2}}(\mathbb{R}^n) \cap B_{q,\theta}^{-\frac{2}{\theta}}(\mathbb{R}^n) \Big) \times \Big(VMO(\mathbb{R}^n) \cap B_{r,\theta}^{1-\frac{2}{\theta}}(\mathbb{R}^n)\Big) \end{align*} for large data time local and \begin{align*} (u_0, \psi_0) \in \Big(L^{\frac{n}{2}}(\mathbb{R}^n) \cap \dot{B}_{q,\theta}^{-\frac{2}{\theta}}(\mathbb{R}^n) \Big) \times \Big(BMO(\mathbb{R}^n) \cap \dot{B}_{r,\theta}^{1-\frac{2}{\theta}}(\mathbb{R}^n)\Big) \ \end{align*} for small data time global, where `…
reviewed by: Debabrata Karmakar
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35K58
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 30H25
Mathematics Subject Classification ID: 30H35
Mathematics Subject Classification ID: 92C17
Mathematics Subject Classification ID: 35K45
zbMATH DE Number: 7451785
zbMATH Keywords: Patlak-Keller-Segel system
zbMATH Keywords: singular limit problem
zbMATH Keywords: scaling invariant spaces
zbMATH Keywords: maximal regularity
zbMATH Keywords: initial layer
zbMATH Keywords: limit of infinite relaxation time
Wikidata QID: Q114216479
MaRDI profile type: Publication
full work available at URL: s42985-021-00134-3
OpenAlex ID: W4200559032
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Maximal regularity and a singular limit problem for the Patlak-Keller-Segel system in the scaling critical space involving \textit{BMO} (DOI)
DOI: 10.1007/S42985-021-00134-3