While studying the weak solutions for incompressible or compressible fluids, they are mainly described by the system of partial differential equations known as Navier-Stokes equations. In most of the literature related to the named equations there are just two equations, namely; law of conservation of mass and momentum given as:
$$\rho_t+\mbox{div}(\rho u)=0 $$ $$ \partial_t(\rho u)+\operatorname{div}(\rho u \otimes u)-\mu\Delta u-\xi\nabla\operatorname{div} u+\nabla(P(\rho))=\rho f $$ while nothing about the energy equations. What is the reason that most of the authors do not including energy equation in their research work. Looking forward for any of your helpful comment. Than you! In specific when the authors are studying the weak solutions as in the work of Eduard Feireisl (On the Existence of Globally Defined Weak Solutions to the Navier–Stokes Equations )1, and like many more