×

zbMATH — the first resource for mathematics

The mathematical analysis of the coupling of a turbulent kinetic energy equation to the Navier-Stokes equation with an eddy viscosity. (English) Zbl 0863.35077
The aim of this paper is to study the system consisting of Navier-Stokes equation and energy equation for an incompressible fluid, considered in a bounded domain in \(\mathbb{R}^n\) \((n=2,3)\), with homogeneous Dirichlet boundary conditions. In the case of bounded eddy viscosity, first, one proves that the system has weak solutions in the steady-state case and in the bidimensional evolution case. Then, existence of weak solutions is obtained when the energy equation is coupled to the three-dimensional evolution Stokes equation. Finally, the regularity of the solutions and the problem of passing to the limit in the equations are considered.
Reviewer: V.A.Sava (Iaşi)

MSC:
35Q30 Navier-Stokes equations
76F05 Isotropic turbulence; homogeneous turbulence
35D05 Existence of generalized solutions of PDE (MSC2000)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Mohammadi, B.; Pironneau, O., Analysis of the k-epsilon model, (1994), Masson Leyden
[2] Launder, B.E.; Spalding, B.D., ()
[3] PIRONNEAU O. Private communication
[4] Lewandowski, R., Modèles de turbulence et équations paraboliques, C. r. acad. sci. Paris, 317, 835-840, (1993) · Zbl 0790.76042
[5] LEWANDOWSKI R., Stabilization of k-Epsilon models for the Navier-Stokes equation (to appear)
[6] Temam, R, ()
[7] Boccardo, L.; Gallouet, T., Nonlinear elliptic and parabolic equations involving measure data, J. funct.analysis, 87, 149-169, (1989) · Zbl 0707.35060
[8] LIONS P.L. & MURAT F., Solutions renormalisées d’équations elliptiques (to appear)
[9] Murat, F., ()
[10] HERBIN R. & GALLOUET T., Preprint Université de Savoie
[11] LEWANDOWSKI R. & MURAT F., In preparation
[12] Diperna, R.J.; Lions, P.L., On the Cauchy problem for the Boltzmann equations: global existence and weak stability, Ann. math., 130, 321-366, (1989) · Zbl 0698.45010
[13] BLANCHARD D., Guibé O. Private communication
[14] CLAIN S., PhD thesis(to appear)
[15] BARANGER J. & MIKELIC A., Stationary solutions to a quasi-Newtonian flow with viscous heating (preprint) · Zbl 0838.76003
[16] Simon, J., Compact sets in the space L^p([0, T], B), Annali mat. pura appl., 146, 4, 65-96, (1987) · Zbl 0629.46031
[17] Lions, J.L., Quelque méthodes de résolutions de problèmes aux limites non linéaires, (1969), Gauthier Villars · Zbl 0189.40603
[18] MURAT F., Private communication
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.