Global Existence of Heat-Conductive Incompressible Viscous Fluids

In this paper, we consider the Cauchy problem of non-stationary motion of heat-conducting incompressible viscous fluids in \(\mathbb{R}^{2}\), where the viscosity and heat-conductivity coefficient vary with the temperature. It is shown that the Cauchy problem has a unique global-in-time strong solution \((u, \theta)(x,t)\) on \(\mathbb{R}^{2}\times(0,\infty)\), provided the initial norm \(\|\nabla u_{0}\|_{L^{2}}\) is suitably small, or the lower-bound of the coefficient of heat conductivity (i.e. \(\underline{\kappa}\)) is large enough, or the derivative of viscosity (i.e. \(|\mu'(\theta)|\)) is small enough.