Archive for Rational Mechanics and Analysis

, Volume 220, Issue 3, pp 937–981 | Cite as

On the Global Regularity of the Two-Dimensional Density Patch for Inhomogeneous Incompressible Viscous Flow

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Abstract

Regarding P.-L. Lions’ open question in Oxford Lecture Series in Mathematics and its Applications, Vol. 3 (1996) concerning the propagation of regularity for the density patch, we establish the global existence of solutions to the two-dimensional inhomogeneous incompressible Navier–Stokes system with initial density given by \({(1 - \eta){\bf 1}_{{\Omega}_{0}} + {\bf 1}_{{\Omega}_{0}^{c}}}\) for some small enough constant \({\eta}\) and some \({W^{k+2,p}}\) domain \({\Omega_{0}}\), with initial vorticity belonging to \({L^{1} \cap L^{p}}\) and with appropriate tangential regularities. Furthermore, we prove that the regularity of the domain \({\Omega_0}\) is preserved by time evolution.

Keywords

Besov Space Inductive Assumption Global Regularity Unique Global Solution Initial Vorticity 
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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Academy of Mathematics and Systems Science, Hua Loo-Keng Key Laboratory of MathematicsThe Chinese Academy of SciencesBeijingChina

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