On the global well-posedness and exponential stability of 3D heat conducting incompressible Navier-Stokes equations with temperature-dependent coefficients and vacuum

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.