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We study global well-posedness of strong solutions for the nonhomogeneousNavier-Stokes equations with density-dependent viscosity and initial densityallowing vanish in $\mathbb{R}^2$. Applying a logarithmic interpolationinequality and delicate energy estimates, we show the global existence of aunique strong solution provided that $\|\nabla\mu(\rho_0)\|_{L^q}$ is suitablysmall, which improves the previous result of Huang and Wang [SIAM J. Math.Anal. 46, 1771--1788 (2014)] to the whole space case. Moreover, we also deriveexponential decay rates of the solution. In particular, there is no need torequire additional initial compatibility condition despite the presence ofvacuum.

Global well-posedness and exponential decay to the Cauchy problem of  nonhomogeneous Navier-Stokes equations with density-dependent viscosity and  vacuum in $\mathbb{R}^2$