The two-dimensional Keller-Segel model after blow-up
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.
scientific article; zbMATH DE number 5604161
scientific article