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

Marius Paicu; Ping Zhang

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arXiv · 1711.04490

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

Marius Paicu·Ping Zhang

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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.

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BibTeX RIS

Marius Paicu, Ping Zhang. 2017-12-09. Striated Regularity of 2-D inhomogeneous incompressible Navier-Stokes system with variable viscosity. https://arxiv.org/abs/1711.04490

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