SUMMARYThis paper is concerned with the equations of non-stationary motion in 3D of heat-conducting incompressible viscous fluids with temperature-dependent viscosity. The conservation of internal energy includes the usual dissipation term. We prove the existence of a 'weak solution with defect measure' to the system of PDEs under consideration. Our method of proof is based on a regularization of the equations of conservation of momentum.

In this paper, we consider weak solutions to the equations of stationary motion of a class of non-Newtonian fluids the constitutive law of which includes the "power law model" as special case. We prove the existence of second order derivatives of weak solutions to these equations. (2000). 35Q30, 76D05, 35J65. Mathematics Subject Classification

This paper is concerned with Kolmogorov's two-equation model for free turbulence in R 3 involving the mean velocity u, the pressure p, an average frequency ω > 0, and a mean turbulent kinetic energy k. We first discuss scaling laws for a slightly more general two-equation models to highlight the special role of the model devised by Kolmogorov in 1942. The main part of the paper consists in proving the existence of weak solutions of Kolmogorov's two-equation model under spaceperiodic boundary conditions in cubes Ω = ( ] 0, l [ ) 3 with l > 0. For this we provide new a priori estimates and invoke existence result for pseudo-monotone operators. * A.M. was supported by Deutsche Forschungsgemeinschaft (DFG) via project A5 within SFB 910.

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