On global regular solutions to the Navier–Stokes equations with heat convection
Volume 108 / 2013
Annales Polonici Mathematici 108 (2013), 155-184
MSC: Primary 76D03; Secondary 76D05, 35Q30.
DOI: 10.4064/ap108-2-3
Abstract
Global existence of regular solutions to the Navier–Stokes equations for velocity and pressure coupled with the heat convection equation for temperature in a cylindrical pipe is shown. We assume the slip boundary conditions for velocity and the Neumann condition for temperature. First we prove long time existence of regular solutions in $[kT,(k+1)T]$. Having $T$ sufficiently large and imposing some decay estimates on $\| f(t)\| _{L_2(\varOmega )}$, $\| f_{,x_3}(t)\| _{L_2(\varOmega )}$ we continue the local solutions step by step up to a global one.
