This paper deals with the 2D incompressible Navier–Stokes equations with density-dependent viscosity over bounded domains. The global existence of strong solutions is established in the vacuum cases, provided the assumption that the bound of density is suitably small, which is in sharp contrast to the recent work (Huang and Wang, 2014), where the smallness assumption on ${\Vert \nabla \mu \left({\rho }_0\right)\Vert }_{L^q}(q>2)$ is needed. Furthermore, we also obtain the exponential decay rates of the spatial gradients of the velocity and the pressure.

中文翻译：

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具有密度依赖粘度系数的二维不可压缩Navier–Stokes方程的整体适定性

本文讨论了在有限域上具有密度依赖粘度的二维不可压缩Navier-Stokes方程。在真空情况下，建立了强解的整体存在性，前提是假设密度的边界适当小，这与最近的工作（Huang和Wang，2014）形成了鲜明的对比，后者的小假设是${\Vert \nabla \mu （{\rho }_0）\Vert }_{{\mathrm{大号}}^q}（q>2）$是必需的。此外，我们还获得了速度和压力的空间梯度的指数衰减率。