A Global Estimate for Regular Axially Symmetric Solutions to the Navier–Stokes Equations Coupled With the Heat Conduction

The axially symmetric solutions to the Navier–Stokes equations coupled with the heat conduction are considered in a bounded cylinder . We assume that vanish on the lateral part of the boundary and vanish on the top and bottom of the cylinder, where we used standard cylindrical coordinates, and is the vorticity of the fluid. Moreover, vanishing of the heat flux through the boundary is imposed. Assuming the existence of a sufficiently regular solution, we derive a global a priori estimate in terms of data. The estimate is such that global regular solutions can be proved. We prove the estimate because some reductions of nonlinearity are found. Moreover, we need that is bounded from below by a positive constant. The quantity is close to 1 for large because .