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We prove long time existence of regular solutions to the Navier-Stokesequations coupled with the heat equation. We consider the system in non-axiallysymmetric cylinder with the slip boundary conditions for the Navier-Stokesequations and the Neumann condition for the heat equation. The long timeexistence is possible because we assumed that derivatives with respect to thevariable along the axis of the cylinder of the initial velocity, initialtemperature and the external force in $L_2$ norms are sufficiently small. Weproved the existence of such solutions that velocity and temperature belong to$W_\sigma^{2,1}(\Omega\times(0,T))$, where $\sigma>{5\over3}$. The existence isproved by the Leray-Schauder fixed point theorem.

Long time existence of regular solutions to 3d Navier-Stokes equations coupled with the heat convection