In this article, we are committed to studying the three-dimensional incompressible Navier-Stokes equations, where the viscosity depends on density according to a power law. We investigate the Cauchy problem by constructing an approximation system and bootstrap argument. Finally, we establish the existence of a global strong solution under the conditions of small initial data and the compatibility condition. Meanwhile, the algebraic decay-in-time rates for the solution are also obtained. It is worth pointing out that the degradation of viscosity is allowed.

This paper focuses on investigating the initial-boundary value problem of incompressible heat conducting Navier-Stokes equations with variable coefficients over bounded domains in ${\mathbb{R}}^3$, where the viscosity coefficient and heat conduction coefficient are powers of temperature. We obtain the global well-posedness of a strong solution under the assumption that the initial data and the measure of the initial vacuum region are sufficiently small. It is worth mentioning that the initial density is allowed to contain vacuum, and there are no restrictions on the power index of the temperature-dependent viscosity coefficient and heat conductivity coefficient. At the same time, the exponential decay-in-time results are also obtained.