This work deals with a system of nonlinear parabolic equations arising in turbulence modelling. The unknowns are the N components of the velocity field u coupled with two scalar quantities θ and φ. The system presents nonlinear turbulent viscosity $A(\theta,\varphi)$ and nonlinear source terms of the form $\theta^2|\nabla u|^2$ and $\theta\varphi|\nabla u|^2$ lying in L1. Some existence results are shown in this paper, including $L^\infty$-estimates and positivity for both θ and φ.