Global Existence of Heat-Conductive Incompressible Viscous Fluids

Abstract

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.