Global well-posedness and blow-up for the 2-D Patlak-Keller-Segel equation
The author proves the existence of mild solutions to the classical Patlak-Keller-Segel system in two dimensions with nonnegative integrable data, together with the global in time existence whenever \(\|n_0\|_{L^1}\leq 8\pi\). The proof is based on the monotonicity formulas for the quantity \(e^{s\Delta}n(x,t)\), and is different from approaches in [\textit{A. Blanchet} et al., Electron. J. Differ. Equ. 2006, Paper No. 44, 33 p. (2006; Zbl 1112.35023); Commun. Pure Appl. Math. 61, No. 10, 1449--1481 (2008; Zbl 1155.35100)] based on the free energy estimates, cf. also the approach for measure initial data in [\textit{P. Biler} and \textit{J. Zienkiewicz}, Bull. Pol. Acad. Sci., Math. 63, No. 1, 41--51 (2015; Zbl 1326.35399)].
scientific article; zbMATH DE number 6810523
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