How does the incompressible Navier-Stokes system read with heat conduction? Where can I find an existence result for its weak solutions?
There is an extensive literature, this could be helpful entry point: Solving Navier-Stokes equations coupled with a heat transfer equation (2015)
In this paper, the dynamics of an incompressible fluid in a bounded connected domain, described by Navier-Stokes equations coupled with a heat transfer equation, is investigated by a method inspired from the non-commutative strategy developed by Bagarello.The solution of systems of partial differential equations is derived with the help of the unbounded self-adjoint densely defined Hamiltonian operator of the physical model and the Hankel transform.
Weak solutions have been studied in On the existence of global weak solutions to the Navier–Stokes equations for viscous compressible and heat conducting fluids (2007). (This paper focuses on compressible flow, but also discusses the literature for incompressible flow.)