The two-dimensional Keller-Segel model after blow-up
type: Item
instance of: scholarly article
title: The two-dimensional Keller-Segel model after blow-up
zbMATH Open document ID: 1180.35131
author: Jean Dolbeault
author: Christian Schmeiser
published in: Discrete and Continuous Dynamical Systems
publication date: 2009-09-16T00:00:00Z
review text: It is now well known that the (nonnegative) solution \(\rho\) to the two-dimensional parabolic-elliptic Keller-Segel system (or Smoluchowski-Poisson equation) \[ \partial_t \rho+\nabla\cdot(\rho\nabla S- \nabla\rho)= 0, \quad x\in\mathbb R^2, \;t>0, \] \[ S(x,t)= -\frac{1}{2\pi} \int_{\mathbb R^2} \log{(|x-y|)}\;\varrho(y,t)\, dy, \] blows up in finite time when the initial condition \(\rho_0\) is such that \(\|\rho_0\|_1>8\pi\). Such a singular behaviour no longer arises if \(S\) is replaced by \[ S_\varepsilon(x,t)= -\frac{1}{2\pi} \int_{\mathbb R^2} \log(|x-y|+\varepsilon) \rho(y,t) \,dy \] for some \(\varepsilon>0\), and the corresponding equation has a global solution \(\rho_\varepsilon\). As \((\rho_\varepsilon)_\varepsilon\) is bounded in \(L^\infty(0,\infty; L^1(\mathbb R^2))\), it has at least a measure-valued (weak) cluster point \(\rho\) (as \(\varepsilon\to 0\)) defined for all times and the question investigated is whether it provides a solution (in a suitable weak sense) to the original equation, in particular after the blowup time. It is shown that \(\rho\) solves a generalized version of the original equation which involves a matrix-valued defect measure \(\nu\) and the support of the trace of \(\nu\) is included in the support of the atomic part of \(\rho\). Assuming further that \(\rho\) can be written as \(\rho=\bar{\rho}+\hat{\rho}\) with \(\bar{\rho}(t)\in L^1(\mathbb R^2)\) and \(\hat{\rho}(t) = \sum_{n=1}^N M_n(t) \delta_{x_n(t)}\) for smooth functions \((M_n)_n\) and \((x_n)_n\), an evolution system is derived for \((M_n,x_n)_n\). Finally, some properties of the profile of \(\rho(t)\) near an atom of \(\rho(t)\) are discussed.
reviewed by: Philippe Laurençot
Mathematics Subject Classification ID: 35B44
Mathematics Subject Classification ID: 92C17
Mathematics Subject Classification ID: 35R09
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 35K59
zbMATH DE Number: 5604161
zbMATH Keywords: measure-valued solution
zbMATH Keywords: continuation after blowup
zbMATH Keywords: defect measure
zbMATH Keywords: Smoluchowski-Poisson equation
Wikidata QID: Q62568236
MaRDI profile type: Publication
full work available at URL: dcds.2009.25.109
OpenAlex ID: W3095900546
DOI: 10.3934/DCDS.2009.25.109
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description: scientific article; zbMATH DE number 5604161