Coupling the Stokes and Navier–Stokes equations with two scalarnonlinear parabolic equations

Abstract

This work deals with a system of nonlinear parabolic equations arisingin turbulence modelling. The unknowns are the N components of the velocityfield u coupled with two scalar quantities θ and φ. The systempresents nonlinear turbulent viscosity $A(\theta,\varphi)$ and nonlinearsource terms of the form $\theta^2|\nabla u|^2$ and $\theta\varphi|\nabla u|^2$ lying in L ¹. Some existence results are shown in this paper, including $L^\infty$ -estimates and positivity for both θ and φ.