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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.

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Authors and Affiliations

  1. Academy of Mathematics and Systems Science, Hua Loo-Keng Key Laboratory of Mathematics, The Chinese Academy of Sciences, Beijing, China

    Xian Liao & Ping Zhang

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  1. Xian Liao
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  2. Ping Zhang
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Correspondence to Ping Zhang.

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Communicated by L. Saint-Raymond

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Liao, X., Zhang, P. On the Global Regularity of the Two-Dimensional Density Patch for Inhomogeneous Incompressible Viscous Flow. Arch Rational Mech Anal 220, 937–981 (2016). https://doi.org/10.1007/s00205-015-0945-z

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  • DOI: https://doi.org/10.1007/s00205-015-0945-z

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