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Global-in-time existence of weak solutions to Kolmogorov’s two-equation model of turbulence. (Sur l’existence globale en temps des solutions faibles pour le modèle de turbulence à deux équations de Kolmogorov.) (English. Abridged French version) Zbl 1320.35262

This article studies the existence of weak solutions \((u,k,w)\) of the Kolmogorov’s model for the turbulent motion of an incompressible fluid in the cube \(\Omega=(]0,a[)^{3}\) (\(0<a<+\infty\)) with periodic boundaries conditions over \(\partial \Omega\): \[ \frac{\partial u}{\partial t}+(u\cdot \nabla)u=\operatorname{div}(\frac{k}{w}D(u))-\nabla p +f,\qquad \operatorname{div}(u)=0, \]
\[ \frac{\partial w}{\partial t}+ u\cdot \nabla w=\operatorname{div}(\frac{k}{w}\nabla w)-w^{2},\qquad \frac{\partial k}{\partial t}+ u\cdot \nabla k=\operatorname{div}(\frac{k}{w}\nabla k)+\frac{k}{w}|D(u)|^{2}-kw, \]
\[ u=u_{0}, k=k_{0}, w=w_{0} \text{ in }\Omega\times \{0\}. \] where \(D(u)=\frac{1}{2}(\nabla u + (\nabla u)^{\perp})\). Here \(u=(u_{1},u_{2}, u_{3})\) denotes the mean velocity, \(p\) is the mean pressure and \(f\in L^2(\Omega\times ]0,T[)\) is a given external force. The function \(k\) represent the mean turbulent kinetic energy and \(w>0\) is the frequency associated with the dissipation of turbulent kinetic energy.
The initial data satisfy the following conditions \(u_{0}\in \overline{C^{\infty}_{\mathrm{per}}(\bar{\Omega})}^{\|\cdot\|_{L^{2}(\Omega)}}\), with \(\operatorname{div}(u_{0})=0\), \(k_{0}\in L^{1}(\Omega)\), \(k_{0}(x)\geq \kappa_{*}>0\) a.e. in \(\Omega\) where \(\kappa_{*}\) is a constant, \(w_{0}\) is measurable in \(\Omega\) and \(w_{*}\leq w(x)\leq w^{*}\) a.e. in \(\Omega\), where \(w_{*}\) and \(w^{*}\) are positive constants.
With all these hypotheses, the authors of this article prove that
\(u\in C_{w}([0,T]; L^{2}(\Omega))\cap L^{2}([0,T]; H^{1}_{\mathrm{per}}(\Omega))\),
\(w\in C_{w}([0,T]; L^{2}(\Omega))\cap L^{2}([0,T]; H^{1}_{\mathrm{per}}(\Omega))\) and \(k\in L^{\infty}([0,T]; L^{1}(\Omega))\).
One of the main arguments of the proof is given by suitable a priori estimates for the functions \(k\) and \(w\): \[ \frac{w_{*}}{1+tw_{*}}\leq w(x,t)\leq \frac{w^{*}}{1+tw^{*}},\text{ and }k(x,t)\geq \frac{k_{}{*}}{1+tw^{*}}, \text{ for a.e. }(x,t)\in \Omega\times]0,T[. \]

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35D30 Weak solutions to PDEs
76F02 Fundamentals of turbulence
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[1] Alexandre, R.; Villani, C., On the Boltzmann equation for long-range interactions, Commun. Pure Appl. Math., 55, 30-70 (2002) · Zbl 1029.82036
[2] Bourbaki, N., Éléments de Mathématique, Livre VI, Intégration 1-4 (1965), Hermann: Hermann, Paris · Zbl 0136.03404
[3] Brézis, H., Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans les Espaces de Hilbert (1973), North-Holland Publ. Comp.: North-Holland Publ. Comp., Amsterdam · Zbl 0252.47055
[4] Droniou, J., Intégration et espaces de Sobolev à valeurs vectorielles
[5] Frisch, U., Turbulence. The Legacy of A.N. Kolmogorov (2004), Cambridge University Press: Cambridge University Press, Cambridge, UK
[6] Harpes, P., Bubbling of approximations for the 2-D Landau-Lifschitz flow, Commun. Partial Differ. Equ., 31, 1-20 (2006) · Zbl 1097.35007
[7] Kolmogorov, A. N., The equations of turbulent motion of an incompressible viscous fluid, Izv. Akad. Nauk SSSR, Ser. Fiz., 6, 56-58 (1942)
[8] Landau, L. D.; Lifschitz, E. M., Lehrbuch der theoretischen Physik. Band VI: Hydromechanik (1991), Akademie-Verlag: Akademie-Verlag, Berlin · Zbl 0997.76501
[9] Lewandowski, R., The mathematical analysis of the coupling of a turbulent kinetic energy equation to the Navier-Stokes equation with an eddy viscosity, Nonlinear Anal., 28, 2, 393-417 (1997) · Zbl 0863.35077
[10] Lin, F.-H.; Liu, C.; Zhang, P., On hydro-dynamics of viscoelastic fluids, Commun. Pure Appl. Math., 58, 1437-1471 (2005) · Zbl 1076.76006
[11] Lions, J.-L., Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires (1969) · Zbl 0189.40603
[12] Naumann, J., An existence theorem for weak solutions to the equations of non-stationary motion of heat-conducting incompressible viscous fluids, J. Nonlinear Convex Anal., 7, 483-497 (2006) · Zbl 1323.76016
[13] Naumann, J., On weak solutions to the equations of non-stationary motion of heat-conducting incompressible viscous fluids: defect measure and energy inequality, Parabolic and Navier-Stokes Equations, vol. 81, 287-296 (2008), Banach Center Publ.: Banach Center Publ., Warsaw, Poland · Zbl 1154.35419
[14] Spalding, D. B., Kolmogorov’s two equation model of turbulence, Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci., 434, 211-216 (1991) · Zbl 0726.76052
[15] Tikhomirov, V. M., Selected Works of A.N. Kolmogorov, vol. 11 (1991), Kluwer Acad. Publishers: Kluwer Acad. Publishers, Dordrecht, The Netherlands
[16] Wilcox, D. C., Turbulence Modeling for CFD (2006), DCW Industries: DCW Industries, La Cañada, CA, USA
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