Weak solutions to incompressible heat-conducting motions with large flux

TL;DR

This paper establishes the existence of weak solutions for the Navier-Stokes and heat equation system in a finite cylinder with large flux, using energy estimates and Galerkin approximation without restrictions on data.

Contribution

It proves the existence of weak solutions for incompressible heat-conducting flows with large flux, employing weighted spaces and energy estimates.

Findings

Existence of weak solutions for large inflow and outflow

Energy estimates valid without restrictions on data

Use of weighted spaces for nonlinear term estimation

Abstract

We analyze the problem for viscous incompressible heat-conduc\-ting fluid in a finite cylinder with large inflow and outflow, modelled with Navier-Stokes equations coupled with the heat equation. We prove energy estimate without restrictions on the magnitudes of the external force, initial data, inflow and outflow. In order to estimate nonlinear terms we use weighted spaces. Next, using Galerkin approximation, we show existence of weak solutions.