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

TL;DR

This paper establishes long-term estimates for solutions to the 3D Navier-Stokes equations coupled with heat convection in a cylindrical domain, without requiring small initial data, but with certain derivative smallness conditions.

Contribution

It provides long-time solution estimates for coupled Navier-Stokes and heat equations without small initial velocity or temperature, under specific derivative smallness conditions.

Findings

Long-term estimates are achieved without small initial velocity or temperature.

Smallness of derivatives with respect to the axial coordinate is necessary.

Results apply to a cylindrical domain with slip and Neumann boundary conditions.

Abstract

We examine the Navier-Stokes equations with homogeneous slip boundary conditions coupled with the heat equation with homogeneous Neumann conditions in a bounded domain in $R^3$. The considered domain is a cylinder with $x_3$-axis. The aim of this paper is to show long time estimates without smallness of the initial velocity, the initial temperature and the external force. To prove the estimate we need however smallness of $L_2$ norms of derivatives with respect to $x_3$ of the initial velocity, the initial temperature and the external force.