Weak solutions to incompressible heat-conducting motions with large flux

We analyze the problem for viscous incompressible heat-conduc\-ting fluid in a finite cylinder with large inflow and outflow, modelled with Navier-Stokes equations coupled with the heat equation. We prove energy estimate without restrictions on the magnitudes of the external force, initial data, inflow and outflow. In order to estimate nonlinear terms we use weighted spaces. Next, using Galerkin approximation, we show existence of weak solutions.