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

TL;DR

This paper proves the long-term existence of smooth solutions to the 3D Navier-Stokes equations coupled with heat convection in a cylindrical domain, under smallness conditions on derivatives along the cylinder axis.

Contribution

It establishes the existence of regular solutions for the coupled system in a non-axially symmetric cylinder with specific boundary conditions, using the Leray-Schauder fixed point theorem.

Findings

Solutions exist for long times under small derivative conditions.

Velocity and temperature belong to specific Sobolev spaces.

Existence proof relies on fixed point theorem.

Abstract

We prove long time existence of regular solutions to the Navier-Stokes equations coupled with the heat equation. We consider the system in non-axially symmetric cylinder with the slip boundary conditions for the Navier-Stokes equations and the Neumann condition for the heat equation. The long time existence is possible because we assumed that derivatives with respect to the variable along the axis of the cylinder of the initial velocity, initial temperature and the external force in $L_2$ norms are sufficiently small. We proved 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 is proved by the Leray-Schauder fixed point theorem.