The global estimate for regular axially-symmetric solutions to the Navier Stokes equations coupled with the heat conduction
Wiesław J. Grygierzec, Wojciech M. Zajączkowski
TL;DR
This work analyzes the 3D incompressible Navier–Stokes system with heat conduction under axial symmetry in a finite cylinder, imposing boundary conditions that eliminate swirl on the side and suppress endcap flux, along with zero heat flux. The authors develop a global a priori bound for regular solutions by introducing a modified stream function $\psi_1=\psi/r$ and reduced vorticity variables $\Phi=\omega_r/r$ and $\Gamma=\omega_\varphi/r$, together with the swirl $u=rv_\varphi$, and then derive a suite of weighted Sobolev, elliptic, and energy estimates connecting these quantities. Critical tools include Hardy-type inequalities, BIN interpolation, and anisotropic Sobolev spaces to handle nonlinearities and time–space coupling; these yield control of $\Phi$, $\Gamma$, and the swirl through data-dependent constants $D_0$–$D_{12}$ and $B_1$. The resulting global a priori bound ensures continuation of local regular solutions in time, advancing the mathematical understanding of regular Navier–Stokes solutions with thermal coupling in cylindrical geometries and providing a framework for further qualitative and quantitative analysis.
Abstract
The axially-symmetric solutions to the Navier-Stokes equations coupled with the heat conduction are considered. in a bounded cylinder $Ω\subset \mathbb{R}^3$. We assume that $v_r, v_{\varphi}, ω_{\varphi}$ vanish on the lateral part $S_1$ of the boundary $\partial Ω$ and $v_z, ω_{\varphi}, \partial_z v_{\varphi}$ vanish on the top and bottom of the cylinder, where we used standard cylindrical coordinates and $ω=\text{rot} v$ 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.