Long time regular solutions to incompressible heat‐conducting motions with large flux

Abstract We consider the Navier–Stokes equations coupled with the heat equation in a cylindrical domain parallel to the ‐axis. On the lateral part of the cylinder, we assume the slip boundary conditions for velocity , and on the top and bottom we have an inflow and outflow. Moreover, we assume a homogeneous Neumann boundary condition for temperature . With smallness of the ‐derivative of initial velocity, initial temperature and the external force field and smallness of derivatives of the inflow and outflow, we prove long‐time estimates for and in Sobolev spaces with finite . Next, by the Leray–Schauder fixed point theorem, we show the existence of solutions to the problem in space . Thanks to the special geometry of , we are able to prove the existence of solutions with large magnitudes of initial velocity, temperature, and inflow‐outflow functions.