On Stability of Solutions to Equations Describing Incompressible Heat-Conducting Motions Under Navier’s Boundary Conditions

Abstract

In this paper we prove existence of global strong-weak two-dimensional solutions to the Navier-Stokes and heat equations coupled by the external force dependent on temperature and the heat dissipation, respectively. The existence is proved in a bounded domain with the Navier boundary conditions for velocity and the Dirichlet boundary condition for temperature. Next, we prove existence of 3d global strong solutions via stability.

Existence of global weak solutions to 3D incompressible heat‐conducting motions with large flux

“…The system is complemented by Navier's boundary conditions and the Dirichlet condition for the temperature, and the problem is considered in a cylinder. The main result of Zadrzyńska and Zaja czkowski 13 is the existence of a global strong-weak solution of three-dimensional problem close to the two-dimensional solution.…”