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

Jacob Bedrossian, Nader Masmoudi

Research output: Contribution to journalArticle

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 (Formula presented). This work improves the small-data results of Biler (Stud Math 114(2):181–192, 1995) and the existence results of Senba and Suzuki (J Funct Anal 191:17–51, 2002). Our work is based on that of Gallagher and Gallay (Math Ann 332:287–327, 2005), 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 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.

Original languageEnglish (US)
Pages (from-to)717-801
Number of pages85
JournalArchive for Rational Mechanics and Analysis
Volume214
Issue number3
DOIs
StatePublished - Oct 17 2014

ASJC Scopus subject areas

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering

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