TY - JOUR
T1 - Existence, Uniqueness and Lipschitz Dependence for Patlak–Keller–Segel and Navier–Stokes in ℝ2 with Measure-Valued Initial Data
AU - Bedrossian, Jacob
AU - Masmoudi, Nader
N1 - Publisher Copyright:
© 2014, Springer-Verlag Berlin Heidelberg.
PY - 2014/10/17
Y1 - 2014/10/17
N2 - 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.
AB - 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.
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U2 - 10.1007/s00205-014-0796-z
DO - 10.1007/s00205-014-0796-z
M3 - Article
AN - SCOPUS:84919874994
SN - 0003-9527
VL - 214
SP - 717
EP - 801
JO - Archive for Rational Mechanics and Analysis
JF - Archive for Rational Mechanics and Analysis
IS - 3
ER -