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arXiv · 1612.08665

Global regularity of 2D density patches for inhomogeneous Navier-Stokes

Francisco Gancedo·Eduardo Garcia-Juarez
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Abstract

This paper is about Lions' open problem on density patches \cite{LIONS}: whether inhomogeneous incompressible Navier-Stokes equations preserve the initial regularity of the free boundary given by density patches. Using classical Sobolev spaces for the velocity, we first establish the propagation of $C^{1+γ}$ regularity with $0<γ<1$ in the case of positive density. Furthermore, we go beyond to show the persistence of a geometrical quantity such as the curvature. In addition, we obtain a proof for $C^{2+γ}$ regularity.

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Francisco Gancedo, Eduardo Garcia-Juarez. 2017-03-31. Global regularity of 2D density patches for inhomogeneous Navier-Stokes. https://arxiv.org/abs/1612.08665

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