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# Communications in Mathematical Sciences

## Volume 12 (2014)

### Number 4

### Global existence of smooth solutions to the $k$-$\epsilon$-model equations for turbulent flows

Pages: 707 – 721

DOI: http://dx.doi.org/10.4310/CMS.2014.v12.n4.a6

#### Authors

#### Abstract

In this paper we are concerned with the global existence of smooth solutions to the $k$-$\epsilon$ model equations for turbulent 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 the $H^3$-framework. The proof relies on energy estimates on velocity, temperature, turbulent kinetic energy, and the 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.

#### Keywords

turbulent flow equations, compressible flows, $k$-$\epsilon$ model equations, classical solution, global existence

#### 2010 Mathematics Subject Classification

35A01, 35Q35, 76F02