Coupling the Stokes and Navier–Stokes equations with two scalar nonlinear parabolic equations

Abstract

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.

Mathematical analysis of some couplings for the Eddington-inspired-Born-Infeld theory of bi-gravity in Bondi coordinates

“…Lemma 3. [ 36] Let X, B and Y be three Banach spaces such that X → B → Y, every imbedding being continuous and the inclusion X → B compact. For 1 p, q < ∞, let W be the Banach space defined as W = {v ∈ L p (X) : dv du ∈ L q (Y)}.…”

“…Remark 7. [36] (a) In (134), the symbol •, • stands for the duality product specified below. (b) We will show that, when f 0 ∈ L ∞ (Ω) (resp.…”

“…Lemma 4. [ 36] Under hypotheses (H1) and (H2), there exists a solution f M of problem (P M ) such that…”

“…Lemma 5. [ 36] Let f M = f M − a. Then, there exists a constant C > 0 (depending on α, a, T, f 0 L 1 (Ω) ) such that for all M > 0…”