Global existence of smooth solutions to three-dimensional turbulent flow equations

In this paper we are concerned with the global existence of smooth solutions to the turbulent flow equations for compressible flows in ${\mathbb{R}}^3$. The global well-posedness is proved under the condition that the initial data are close to the standard equilibrium state in $H^3$-framework. The proof relies on energy estimates about velocity, temperature, turbulent kinetic energy and rate of viscous dissipation. We use several new techniques to overcome the difficulties from the product of two functions and higher order norms. This is the first result concerning $k$-$\epsilon$ model equations.