Global Regularity of 2D Density Patches for Inhomogeneous Navier–Stokes

Abstract

This paper is about Lions’ open problem on density patches (Lions in Mathematical topics in fluid mechanics. Vol. 1, volume 3 of Oxford Lecture series in mathematics and its applications, Clarendon Press, Oxford University Press, New York, 1996): whether or not 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 $${C^{1+\gamma}}$$ regularity with $${0 < \gamma < 1}$$ in the case of positive density. Furthermore, we go beyond this to show the persistence of a geometrical quantity such as the curvature. In addition, we obtain a proof for $${C^{2+\gamma}}$$ regularity.