# Article

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Keywords:
large data existence; suitable weak solution; Navier-Stokes-Fourier equations; incompressible fluid; the viscosity increasing with a scalar quantity; regularity; turbulent kinetic energy model
Summary:
In this paper, we establish the large-data and long-time existence of a suitable weak solution to an initial and boundary value problem driven by a system of partial differential equations consisting of the Navier-Stokes equations with the viscosity $\nu$ polynomially increasing with a scalar quantity $k$ that evolves according to an evolutionary convection diffusion equation with the right hand side $\nu (k)|{\pmb{\mathsf{D}}}(\vec{v})|^2$ that is merely $L^1$-integrable over space and time. We also formulate a conjecture concerning regularity of such a solution.
References:
[1] Bernardi C., Chacón Rebollo T., Gómez Mármol M., Lewandowski R., Murat F.: A model for two coupled turbulent fluids, III. Numerical approximation by finite elements. Numer. Math. 98 (2004), no. 1, 33–66. DOI 10.1007/s00211-003-0490-9 | MR 2076053
[2] Bernardi C., Chacón Rebollo T., Hecht F., Lewandowski R.: Automatic insertion of a turbulence model in the finite element discretization of the Navier-Stokes equations. Math. Models Methods Appl. Sci. 19 (2009), no. 7, 1139–1183. DOI 10.1142/S0218202509003747 | MR 2553180
[3] Bernardi C., Chacón Rebollo T., Lewandowski R., Murat F.: A model for two coupled turbulent fluids, I. Analysis of the system. Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, Vol. XIV (Paris, 1997/1998), Stud. Math. Appl., 31, North-Holland, Amsterdam, 2002, pp. 69–102. DOI 10.1016/S0168-2024(02)80006-6 | MR 1935990
[4] Bernardi C., Chacón Rebollo T., Lewandowski R., Murat F.: A model for two coupled turbulent fluids, II. Numerical analysis of a spectral discretization. SIAM J. Numer. Anal. 40 (2002), no. 6, 2368–2394 (electronic) (2003). DOI 10.1137/S0036142901385829 | MR 1974191
[5] Blanke B., Delecluse P.: Variability of the Tropical Atlantic Ocean simulated by a general circulation model with two different mixed-layer physics. J. Phys. Oceanogr. 23 (1993), 1363–1388. DOI 10.1175/1520-0485(1993)023<1363:VOTTAO>2.0.CO;2
[6] Brossier F., Lewandowski R.: Impact of the variations of the mixing length in a first order turbulent closure system. M2AN Math. Model. Numer. Anal. 36 (2002), no. 2, 345–372. DOI 10.1051/m2an:2002016 | MR 1906822 | Zbl 1040.35057
[7] Bulíček M., Málek J., Rajagopal K.R.: Mathematical analysis of unsteady flows of fluids with pressure, shear-rate, and temperature dependent material moduli that slip at solid boundaries. SIAM J. Math. Anal.41 (2009), no. 2, 665–707. DOI 10.1137/07069540X | MR 2515781
[8] Bulíček M., Feireisl E., Málek J.: A Navier-Stokes-Fourier system for incompressible fluids with temperature dependent material coefficients. Nonlinear Anal. Real World Appl. 10 (2009), no. 2, 992–1015. MR 2474275
[9] Bulíček M., Málek J., Rajagopal K.R.: Navier's slip and evolutionary Navier-Stokes like systems with pressure and shear-rate dependent viscosity. Indiana Univ. Math. J. 56 (2007), 51–85. DOI 10.1512/iumj.2007.56.2997 | MR 2305930 | Zbl 1129.35055
[10] Caffarelli L., Kohn R., Nirenberg L.: Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35 (1982), no. 6, 771–831. DOI 10.1002/cpa.3160350604 | MR 0673830 | Zbl 0509.35067
[11] Chacon T., Pironneau O.: On the mathematical foundations of the $k$-$\epsilon$ turbulent model. Vistas in Applied Mathematics, Transl. Ser. Math. Engrg., Optimization Software, New York, 1986, pp. 44–56. MR 0859923 | Zbl 0618.76049
[12] Chácon Rebollo T.: Oscillations due to the transport of microstructures. SIAM J. Appl. Math. 48 (1988), no. 5, 1128–1146. DOI 10.1137/0148067 | MR 0960475
[13] Feireisl E., Málek J.: On the Navier-Stokes equations with temperature-dependent transport coefficients. Differ. Equ. Nonlinear Mech. 2006, Art. ID 90616, 14 pp. (electronic). MR 2233755
[14] Gallouët T., Lederer J., Lewandowski R., Murat F., Tartar L.: On a turbulent system with unbounded eddy viscosities. Nonlinear Anal. 52 (2003), no. 4, 1051–1068. DOI 10.1016/S0362-546X(01)00890-2 | MR 1941245
[15] Kolmogorov A.N.: Equations of turbulent motion in an incompressible fluid. Izv. Akad. Nauk SSSR, Seria fizicheska 6 (1942), no. 1–2, 56–58.
[16] Kolmogorov A.N.: Selected works of A.N. Kolmogorov, Vol. I. Mathematics and Mechanics. With commentaries by V.I. Arnol'd, V.A. Skvortsov, P.L. Ul'yanov et al., translated from the Russian original by V.M. Volosov. Edited and with a preface, foreword and brief biography by V.M. Tikhomirov. Mathematics and its Applications (Soviet Series), 25, Kluwer Academic Publishers Group, Dordrecht, 1991. MR 1175399
[17] Launder B.E., Spalding D.B.: Mathematical Models of Turbulence. Academic Press, 1972. Zbl 0288.76027
[18] Lederer J., Lewandowski R.: A RANS 3D model with unbounded eddy viscosities. Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), no. 3, 413–441. DOI 10.1016/j.anihpc.2006.03.011 | MR 2321200 | Zbl 1132.35069
[19] Lewandowski R.: Les équations de Stokes et de Navier-Stokes couplées avec l'équation de l'énergie cinétique turbulente. C.R. Acad. Sci. Paris Sér. I Math. 318 (1994), no. 12, 1097–1102. MR 1282351 | Zbl 0806.35138
[20] Lewandowski R.: Analyse Mathématique et Océanographie. Masson, 1997.
[21] 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 (1997), no. 2, 393–417. DOI 10.1016/0362-546X(95)00149-P | MR 1418142 | Zbl 0863.35077
[22] Lewandowski R., Pichot G.: Numerical simulation of water flow around a rigid fishing net. Comput. Methods Appl. Mech. Engrg. 196 (2007). no. 45–48, 4737–4754. DOI 10.1016/j.cma.2007.06.007 | MR 2354460 | Zbl 1173.76410
[23] Lions P.L.: Mathematical Topics in Fluid Mechanics, Vol. 1. Oxford Lecture Series in Mathematics and its Applications, 3, The Clarendon Press, Oxford University Press, New York, 1996. MR 1422251 | Zbl 0866.76002
[24] Málek J., Nečas J., Rokyta M., Růžička M.: Weak and measure-valued solutions to evolutionary PDEs. Chapman & Hall, London, 1996. MR 1409366
[25] McLaughlin D.W., Papanicolaou G.C., Pironneau O.R.: Convection of microstructure and related problems. SIAM J. Appl. Math. 45 (1985), no. 5, 780–797. DOI 10.1137/0145046 | MR 0804006 | Zbl 0622.76062
[26] Mohammadi B., Pironneau O.: Analysis of the $k$-epsilon turbulence model. RAM: Research in Applied Mathematics, Masson, Paris, 1994. MR 1296252
[27] Naumann J.: On the existence of weak solutions to the equations of non-stationary motion of heat-conducting incompressible viscous fluids. Math. Methods Appl. Sci. 29 (2006), no. 16, 1883–1906. DOI 10.1002/mma.754 | MR 2259989 | Zbl 1106.76016
[28] Pichot G., Germain G., Priour D.: On the experimental study of the flow around a fishing net. European Journal of Mechanics - B/Fluids, 28 (2009), 103–116. DOI 10.1016/j.euromechflu.2008.02.002 | Zbl 1153.76309
[29] Simon J.: Compact sets in the space $L^p(0,T;B)$. Ann. Mat. Pura Appl. (4) 146 (1987), 65–96. MR 0916688
[30] Spalding D.B.: Kolmogorov's two-equation model of turbulence. Turbulence and stochastic processes: Kolmogorov's ideas 50 years on. Proc. Roy. Soc. London Ser. A 434 (1991), no. 1890, 211–216. MR 1124931
[31] Vasseur A.: A new proof of partial regularity of solutions to Navier-Stokes equations. Nonlinear Differ. Equ. Appl. 14 (2007), 753–785. DOI 10.1007/s00030-007-6001-4 | MR 2374209 | Zbl 1142.35066
[32] Vialard J., Delecluse P.: An OGCM study for the TOGA decade, Part II: Barrier-layer formation and variability. J. Phys. Oceanogr. 28 (1998), 1089–1106. DOI 10.1175/1520-0485(1998)028<1089:AOSFTT>2.0.CO;2

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