Striated Regularity of 2-D Inhomogeneous Incompressible Navier-Stokes System with Variable Viscosity

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 L1 in time Lipschitz estimate of the velocity field can not be obtained by energy method (see Danchin and Mucha in The incompressible Navier-Stokes equations in vacuum; Liao and Zhang in Arch Ration Mech Anal 220:937-981, 2016; Commun Pure Appl Math 72:835-884, 2019 for instance). Motivated by the key idea of Chemin to solve 2-D vortex patch of ideal fluid (Chemin in Invent Math 103:599-629, 1991; Ann Sci ecole Norm Sup 26(4):517-542, 1993), namely, striated regularity can help to get the L infinity boundedness of the double Riesz transform, we derive the a prioriL1 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 H52 regularity of the interface between fluids with different densities and viscosities.