We study some nonlinear elliptic systems governing the steady-state of a two-equation turbulence model that has been derived from the so-called k-ε model. Two kinds of problems are considered: in the first one, we drop out transport terms and we deduce the existence of a solution for N ≥ 2; in the second one we take into account all transport terms; in this case, the existence result holds for N = 2 or 3. Positivity and L ∞ -regularity of the scalar quantities are also shown here.
Abstract. 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(θ, ϕ) and nonlinear source terms of the form θ 2 |∇u| 2 and θϕ|∇u| 2 lying in L 1 . Some existence results are shown in this paper, including L ∞ -estimates and positivity for both θ and ϕ.Résumé. Nousétudions un système non-linéaire d'équations du type parabolique provenant de la modélisation de la turbulence. Les inconnues sont les N composantes du champ des vitesses u couplées avec deux grandeurs scalaires θ et ϕ. Ce système présente un terme de diffusion non-linéaire sous forme matricielle A(θ, ϕ) et les termes sources non-linéaires θ 2 |∇u| 2 et θϕ|∇u| 2 appartenantà L 1 . On démontre alors quelques résultats d'existence de solutions, ainsi que des estimations dans L ∞ et positivité pour θ et ϕ.
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