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

Abstract

This paper is about Lions’ open problem on density patches: 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 C1+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 C2+γ regularity.