The 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 $\mathrm{\Omega }\subset {\mathbb{R}}^3$. We assume that $v_r,v_{\phi },{\omega }_{\phi }$ vanish on the lateral part $S_1$ of the boundary $\mathrm{\partial }\mathrm{\Omega }$ and $v_z,{\omega }_{\phi },{\mathrm{\partial }}_z{v}_{\phi }$ vanish on the top and bottom of the cylinder, where we used standard cylindrical coordinates and $\omega =\text{rot}v$ is the vorticity of the fluid. Moreover, vanishing of the heat flux through the boundary is imposed. Assuming existence of a sufficiently regular solution we derive a global a priori estimate in terms of data. The estimate is such that a global regular solutions can be proved. We prove the estimate because some reduction of nonlinearity are found.Moreover, deriving the global estimate for a local solution implies a possibility of its extension in time as long as the estimate holds.