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

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 $C^{1+\gamma }$ regularity with $0<\gamma <1$ 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 $C^{2+\gamma }$ regularity.

Some typos corrected, added references and comments about free boundary conditions, introduction expanded, 22 pages

References in corpus (2)

• On the persistence of Hölder regular patches of density for the inhomogeneous Navier-Stokes equations
• Global regularity for 2D Boussinesq temperature patches with no diffusion

Cited by in corpus (2)

• The incompressible navier-stokes equations in vacuum
• Striated Regularity of 2-D inhomogeneous incompressible Navier-Stokes system with variable viscosity