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Nonlinear Analysis

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

Ping Zhang, Institute of Mathematics, AMSS, Chinese Academy of Sciences

Location:  Hill 705
Date & time: Tuesday, 20 February 2018 at 1:40PM - 2:40PM

Abstract: In this talk, 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. Motivated by the key idea of Chemin to solve 2-D vortex patch of ideal fluid, 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.