Striated Regularity of 2-D inhomogeneous incompressible Navier-Stokes system with variable viscosity

TL;DR

This paper proves the global existence and uniqueness of strong solutions for 2D inhomogeneous Navier-Stokes equations with variable viscosity, demonstrating the propagation of interface regularity despite initial density discontinuities.

Contribution

It introduces a novel approach using striated regularity to obtain Lipschitz estimates for velocity, extending results to variable viscosity cases with discontinuous density interfaces.

Findings

Established global strong solutions for 2D inhomogeneous Navier-Stokes with variable viscosity.

Proved propagation of interface regularity in density discontinuities.

Extended prior results by overcoming challenges in Lipschitz estimates without energy methods.

Abstract

In this paper, we investigate the global existence and uniqueness of strong solutions to 2D incompressible inhomogeneous Navier-Stokes equations with viscous coefficient depending on the density and with initial density being discontinuous across some smooth interface. Compared with the previous results for the inhomogeneous Navier-Stokes equations with constant viscosity, the main difficulty here lies in the fact that the $L^1$ in time Lipschitz estimate of the velocity field can not be obtained by energy method (see \cite{DM17,LZ1, LZ2} for instance). Motivated by the key idea of Chemin to solve 2-D vortex patch of ideal fluid (\cite{Chemin91, Chemin93}), namely, striated regularity can help to get the $L^\infty$ boundedness of the double Riesz transform, we derive the {\it a priori} $L^1$ in time Lipschitz estimate of the velocity field under the assumption that the viscous coefficient is close enough to a positive constant in the bounded function space. As an application, we shall prove the propagation of $H^3$ regularity of the interface between fluids with different densities.