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We establish a new local well-posedness result in the space of finite Borelmeasures for mild solutions of the parabolic-elliptic Patlak-Keller-Segel (PKS)model of chemotactic aggregation in two dimensions. Our result only requiresthat the initial measure satisfy the necessary assumption $\max_{x \in\Real^2}\mu(\set{x}) < 8\pi$. This work improves the small-data results ofBiler and the existence results of Senba and Suzuki. Our work is based on thatof Gallagher and Gallay, who prove the uniqueness and log-Lipschitz continuityof the solution map for the 2D Navier-Stokes equations (NSE) withmeasure-valued initial vorticity. We refine their techniques and present analternative version of their proof which yields existence, uniqueness and\emph{Lipschitz} continuity of the solution maps of both PKS and NSE. Manysteps are more difficult for PKS than for NSE, particularly on the level of thelinear estimates related to the self-similar spreading solutions.

Existence, Uniqueness and Lipschitz Dependence for Patlak-Keller-Segel and Navier-Stokes in R2 with Measure-valued Initial Data