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On the global well-posedness of 2-D density-dependent Navier-Stokes system with variable viscosity (1301.2371v1)

Published 11 Jan 2013 in math.AP

Abstract: Given solenoidal vector $u_0\in H{-2\d}\cap H1(\R2),$ $\r_0-1\in L2(\R2),$ and $\r_0 \in L\infty\cap\dot{W}{1,r}(\R2)$ with a positive lower bound for $\d\in (0,\f12)$ and $2<r<\f{2}{1-2\d},$ we prove that 2-D incompressible inhomogeneous Navier-Stokes system \eqref{1.1} has a unique global solution provided that the viscous coefficient $\mu(\r_0)$ is close enough to 1 in the $L\infty$ norm compared to the size of $\d$ and the norms of the initial data. With smoother initial data, we can prove the propagation of regularities for such solutions. Furthermore, for $1<p<4,$ if $(\r_0-1,u_0)$ belongs to the critical Besov spaces $\dB{\f2p}_{p,1}(\R2)\times \bigl(\dB{-1+\f2p}_{p,1}\cap L2(\R2)\bigr)$ and the $\dB{\f2p}_{p,1}(\R2)$ norm of $\r_0-1$ is sufficiently small compared to the exponential of $|u_0|{L2}2+|u_0|{\dB{-1+\f2p}_{p,1}},$ we prove the global well-posedness of \eqref{1.1} in the scaling invariant spaces. Finally for initial data in the almost critical Besov spaces, we prove the global well-posedness of \eqref{1.1} under the assumption that the $L\infty$ norm of $\r_0-1$ is sufficiently small.

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