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

Abstrakt

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 11/5, 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 p > 5/2.

The main novelty is that we do not renormalize the temperature equation, but rather construct a solution which fulfils the entropy equality.

Klíčová slova

• incompressible fluid
• heat conducting fluid
• stability
• inhomogeneous temperature
• non-Newtonian fluid
• entropy equality
• renormalized solution