## The mathematical analysis of the coupling of a turbulent kinetic energy equation to the Navier-Stokes equation with an eddy viscosity.(English)Zbl 0863.35077

The aim of this paper is to study the system consisting of Navier-Stokes equation and energy equation for an incompressible fluid, considered in a bounded domain in $$\mathbb{R}^n$$ $$(n=2,3)$$, with homogeneous Dirichlet boundary conditions. In the case of bounded eddy viscosity, first, one proves that the system has weak solutions in the steady-state case and in the bidimensional evolution case. Then, existence of weak solutions is obtained when the energy equation is coupled to the three-dimensional evolution Stokes equation. Finally, the regularity of the solutions and the problem of passing to the limit in the equations are considered.
Reviewer: V.A.Sava (Iaşi)

### MSC:

 35Q30 Navier-Stokes equations 76F05 Isotropic turbulence; homogeneous turbulence 35D05 Existence of generalized solutions of PDE (MSC2000)
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### References:

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