We consider the global existence and large-time asymptotic behavior of strong solutions to the Cauchy problem of the three-dimensional nonhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity and vacuum. We establish some key a priori exponential decay-in-time rates of the strong solutions. Then after using these estimates, we also obtain the global existence of strong solutions in the whole three-dimensional space, provided that the initial velocity is suitably small in the $\dot H^\beta$-norm for some $\beta\in(1/2,1].$ Note that this result is proved without any smallness conditions on the initial density. Moreover, the density can contain vacuum states and even have compact support initially.

中文翻译：

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3D Navier-Stokes 方程在无界域中具有与密度相关的粘度和真空的全局适定性和指数稳定性

我们考虑了具有密度依赖的粘度和真空的三维非齐次不可压缩 Navier-Stokes 方程的柯西问题的强解的全局存在性和大时间渐近行为。我们建立了强解的一些关键的先验指数随时间衰减率。然后在使用这些估计之后，我们还获得了在整个三维空间中强解的全局存在性，前提是初始速度在 $\dot H^\beta$-norm 中对于某些 $\beta\in (1/2,1].$ 注意这个结果是在初始密度没有任何小条件的情况下证明的。而且，密度可以包含真空状态，甚至最初具有紧支撑。