How does the incompressible Navier-Stokes system read with heat conduction?
Where can I find an existence result for its weak solutions?



Incompressible Navier-Stokes equation with heat conduction
There is an extensive literature, this could be helpful entry point:  [Solving Navier-Stokes equations coupled with a heat transfer equation](https://arxiv.org/abs/1509.00820) (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](https://core.ac.uk/download/pdf/82480680.pdf) (2007). (This paper focuses on compressible flow, but also discusses the literature for incompressible flow.)


