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

Abstract

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