On solutions for a generalized Navier–Stokes–Fourier system fulfilling the entropy equality

We consider a flow of a non-Newtonian heat conducting incompressible fluid in a bounded domain subjected to the homogeneous Dirichlet boundary condition for the velocity field and the Dirichlet boundary condition for the temperature. In three dimensions, for a power-law index greater or equal to [Formula: see text] , we show the existence of a solution fulfilling the entropy equality. The entropy equality can be formally deduced from the energy equality by renormalization. However, such a procedure can be justified by the DiPerna–Lions theory only for [Formula: see text]. The main novelty is that we do not renormalize the temperature equation, but rather construct a solution which fulfils the entropy equality. This article is part of the theme issue ‘Non-smooth variational problems and applications’.