## On weak solutions to the equations of non-stationary motion of heat-conducting incompressible viscous fluids: defect measure and energy equality

### Volume 81 / 2008

Banach Center Publications 81 (2008), 287-296
MSC: Primary 35Q30; Secondary 76D05.
DOI: 10.4064/bc81-0-19

#### Abstract

We consider the non-stationary Navier-Stokes equations completed by the equation of conservation of internal energy. The viscosity of the fluid is assumed to depend on the temperature, and the dissipation term is the only heat source in the conservation of internal energy. For the system of PDE's under consideration, we prove the existence of a weak solution such that: 1) the weak form of the conservation of internal energy involves a defect measure, and 2) the equality for the total energy is satisfied.