On Existence and Regularity of Solutions for a Stationary Navier-Stokes System Coupled to an Equation for the Turbulent Kinetic Energy.

Pierre-Etienne Druet

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976),

MSC 2000

35D05 Existence of generalized solutions

35J60 Nonlinear PDE of elliptic type

76D05 Navier-Stokes equations

76F60 $k$-$\varepsilon$ modeling

Abstract
We investigate a stationary model for turbulent flows, in which the Navier-Stokes system is coupled to an equation for the density of turbulent kinetic energy through a bounded coefficient of eddy viscosity. We prove the existence of weak solutions for which the gradients are higher integrable. The latter property allows us to prove that the model is well posed in the two dimensional case, provided that the external forcing remains sufficiently small.