On the convergence of statistical solutions of the 3D Navier-Stokes-\alpha model as \alpha vanishes

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January 2014, 34(1): 19-49. doi: 10.3934/dcds.2014.34.19

On the convergence of statistical solutions of the 3D Navier-Stokes-$\alpha$ model as $\alpha$ vanishes

Anne Bronzi ^1, and Ricardo Rosa ^2,

1.	Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, Ilha do Fundão, Rio de Janeiro, RJ, 21941-909, Brazil
2.	Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, Ilha do Fundão, Rio de Janeiro, RJ 21941-909

Received October 2012 Revised April 2013 Published June 2013

Abstract
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In this paper statistical solutions of the 3D Navier-Stokes-$\alpha$ model with periodic boundary condition are considered. It is proved that under certain natural conditions statistical solutions of the 3D Navier-Stokes-$\alpha$ model converge to statistical solutions of the exact 3D Navier-Stokes equations as $\alpha$ goes to zero. The statistical solutions considered here arise as families of time-projections of measures on suitable trajectory spaces.

Keywords: Statistical solutions, Navier-Stokes-$\alpha$ model., Navier-Stokes equations.

Mathematics Subject Classification: Primary: 76D06, 35Q30, 76D05; Secondary: 35Q3.

Citation: Anne Bronzi, Ricardo Rosa. On the convergence of statistical solutions of the 3D Navier-Stokes-$\alpha$ model as $\alpha$ vanishes. Discrete and Continuous Dynamical Systems, 2014, 34 (1) : 19-49. doi: 10.3934/dcds.2014.34.19

References:

[1]	C. D. Aliprantis and K. C. Border, "Infinite Dimensional Analysis. A Hitchhiker's Guide,'' Third edition, Springer, Berlin, 2006.
[2]	G. K. Batchelor, "The Theory of Homogeneous Turbulence,'' Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, 1953.
[3]	H. Bercovici, P. Constantin, C. Foias and O. P. Manley, Exponential decay of the power spectrum of turbulence, J. Stat. Phys., 80 (1995), 579-602. doi: 10.1007/BF02178549.
[4]	A. Bronzi, C. Mondaini and R. Rosa, On the convergence of statistical solutions, work in progress.
[5]	M. Capinski and N. Cutland, A simple proof of existence of weak and statistical solutions of Navier-Stokes equations, Proc. Roy. Soc. London Ser. A, 436 (1992), 1-11. doi: 10.1098/rspa.1992.0001.
[6]	S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, Camassa-Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett., 81 (1998), 5338-5341. doi: 10.1103/PhysRevLett.81.5338.
[7]	S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in channels and pipes, Phys. Fluids, 11 (1999), 2343-2353. doi: 10.1063/1.870096.
[8]	S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, The Camassa-Holm equations and turbulence, Phys. D, 133 (1999), 49-65. doi: 10.1016/S0167-2789(99)00098-6.
[9]	P. Constantin and C. Foias, "Navier-Stokes Equations,'' Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.
[10]	P. Constantin, C. Foias and O. P. Manley, Effects of the forcing function on the energy spectrum in 2-D turbulence, Phys. Fluids, 6 (1994), 427-429. doi: 10.1063/1.868042.
[11]	P. Constantin and F. Ramos, Inviscid limit for damped and driven incompressible Navier-Stokes equations in $\mathbb{R}^2$, Comm. Math. Phys., 275 (2007), 529-551. doi: 10.1007/s00220-007-0310-7.
[12]	P. Constantin and J. Wu, Statistical solutions of the Navier-Stokes equations on the phase space of vorticity and the inviscid limits, J. Math. Phys., 38 (1997), 3031-3045. doi: 10.1063/1.532032.
[13]	F. Flandoli, An introduction to 3D stochastic fluid dynamics, in "SPDE in hydrodynamic: Recent progress and prospects,'' Lecture Notes in Math., 1942, Springer, Berlin, (2008), 51-150 doi: 10.1007/978-3-540-78493-7_2.
[14]	C. Foias, Statistical study of Navier-Stokes equations. I, Rend. Sem. Mat. Univ. Padova, 48 (1972), 219-348.
[15]	C. Foias, Statistical study of Navier-Stokes equations. II, Rend. Sem. Mat. Univ. Padova, 49 (1973), 9-123.
[16]	C. Foias, A functional approach to turbulence, Russian Math. Survey, 29 (1974), 293-326.
[17]	C. Foias, What do the Navier-Stokes equations tell us about turbulence?, in "Harmonic Analysis and Nonlinear Differential Equations'' (Riverside, CA, 1995), Contemp. Math., 208, Amer. Math. Soc., Providence, RI, (1997), 151-180. doi: 10.1090/conm/208/02739.
[18]	C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence. Advances in nonlinear mathematics and science, Phys. D, 152/153 (2001), 505-519. doi: 10.1016/S0167-2789(01)00191-9.
[19]	C. Foias, D. D. Holm and E. S. Titi, The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory, J. Dynam. Differential Equations, 14 (2002), 1-35. doi: 10.1023/A:1012984210582.
[20]	C. Foias, O. P. Manley, R. M. S. Rosa and R. Temam, Estimates for the energy cascade in three-dimensional turbulent flows, Comptes Rendus Acad. Sci. Paris, Série I, 333 (2001), 499-504. doi: 10.1016/S0764-4442(01)02008-0.
[21]	C. Foias, O. Manley, R. Rosa and R. Temam, "Navier-Stokes Equations and Turbulence,'' Encyclopedia of Mathematics and its Applications, Vol. 83, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511546754.
[22]	C. Foias and G. Prodi, Sur les solutions statistiques des équations de Navier-Stokes, Ann. Mat. Pura Appl. (4), 111 (1976), 307-330. doi: 10.1007/BF02411822.
[23]	C. Foias, R. M. S. Rosa and R. Temam, A note on statistical solutions of the three-dimensional Navier-Stokes equations: The time-dependent case, Comptes Rendus Acad. Sci. Paris, 348 (2010), 235-240. doi: 10.1016/j.crma.2009.12.017.
[24]	C. Foias, R. M. S. Rosa and R. Temam, A note on statistical solutions of the three-dimensional Navier-Stokes equations: The stationary case, Comptes Rendus Acad. Sci. Paris, 348 (2010), 347-353. doi: 10.1016/j.crma.2009.12.018.
[25]	C. Foias, R. Rosa and R. Temam, Properties of time-dependent statistical solutions of the three-dimensional Navier-Stokes equations, to appear in Annales de l'Institut Fourier, arXiv:1111.6257v1.
[26]	C. Foias, R. Rosa and R. Temam, Properties of stationary statistical solutions of the three-dimensional Navier-Stokes equations, in preparation.
[27]	A. V. Fursikov, The closure problem for the Friedman-Keller infinite chain of moment equations, corresponding to the Navier-Stokes system, in "Fundamental Problematic Issues in Turbulence" (Monte Verita, 1998), Trends Math., Birkhäuser, Basel, (1999), 17-24.
[28]	J. O. Hinze, "Turbulence: An Introduction to its Mechanism and Theory,'' Second edition, McGraw-Hill Book Co., Inc., New York, New York, 1975.
[29]	M. Holst, E. Lunasin and G. Tsogtgerel, Analysis of a general family of regularized Navier-Stokes and MHD models, J. Nonlinear Sci., 20 (2010), 523-567. doi: 10.1007/s00332-010-9066-x.
[30]	E. Hopf, Statistical hydromechanics and functional calculus, J. Rational Mech. Anal., 1 (1952), 87-123.
[31]	O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'' Revised English edition, Translated from the Russian by Richard A. Silverman, Gordon and Breach Science Publishers, New York-London, 1963.
[32]	A. S. Monin and A. M. Yaglom, "Statistical Fluid Mechanics: Mechanics of Turbulence,'' Volume 2, MIT Press, Cambridge, MA, 1975.
[33]	Yu. V. Prohorov, Convergence of random processes and limit theorems in probability theory, Teor. Veroyatnost. i Primenen., 1 (1956), 177-238.
[34]	F. Ramos, R. Rosa and R. Temam, Statistical estimates for channel flows driven by a pressure gradient, Phys. D, 237 (2008), 1368-1387. doi: 10.1016/j.physd.2008.03.013.
[35]	F. Ramos and E. S. Titi, Invariant measures for the 3D Navier-Stokes-Voigt equations and their Navier-Stokes limit, Discrete Contin. Dyn. Syst., 28 (2010), 375-403. doi: 10.3934/dcds.2010.28.375.
[36]	R. M. S. Rosa, Some results on the Navier-Stokes equations in connection with the statistical theory of stationary turbulence, Appl. Math., 47 (2002), 485-516. doi: 10.1023/A:1023297721804.
[37]	R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,'' Third edition, Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam-New York, (1984); Reedition in the AMS Chelsea series, AMS, Providence, 2001.
[38]	R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis,'' Second edition, CBMS-NSF Regional Conference Series in Applied Mathematics, 66, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. doi: 10.1137/1.9781611970050.
[39]	H. Tennekes and J. L. Lumley, "A First Course in Turbulence,'' MIT Press, Cambridge, Mass., 1972.
[40]	F. Topsøe, Compactness in spaces of measures, Studia Math., 36 (1970), 195-222.
[41]	F. Topsøe, "Topology and Measure,'' Lecture Notes in Mathematics, Vol. 133, Springer-Verlag, Berlin-New York, 1970.
[42]	F. Topsøe, Compactness and tightness in a space of measures with the topology of weak convergence, Math. Scand., 34 (1974), 187-210.
[43]	M. I. Višik and A. V. Foursikov, L'équation de Hopf, les solutions statistiques, les moments correspondant aux systèmes des équations paraboliques quasilinéaires, J. Math. Pures Appl. (9), 56 (1977), 85-122.
[44]	M. I. Višik and A. V. Fursikov, Translationally homogeneous statistical solutions and individual solutions with infinite energy of a system of Navier-Stokes equations, (Russian) Sibirsk. Mat. Zh., 19 (1978), 1005-1031, 1213.
[45]	M. I. Višik and A. Fursikov, "Mathematical Problems of Statistical Hydrodynamics,'' Kluwer, Dordrecht, 1988.
[46]	M. I. Višik, E. S. Titi and V. V. Chepyzhov, On the convergence of trajectory attractors of the three-dimensional Navier-Stokes $\alpha$-model as $\alpha \rightarrow 0$, Sb. Math., 198 (2007), 1703-1736. doi: 10.1070/SM2007v198n12ABEH003902.

show all references

References:

[1]	C. D. Aliprantis and K. C. Border, "Infinite Dimensional Analysis. A Hitchhiker's Guide,'' Third edition, Springer, Berlin, 2006.
[2]	G. K. Batchelor, "The Theory of Homogeneous Turbulence,'' Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, 1953.
[3]	H. Bercovici, P. Constantin, C. Foias and O. P. Manley, Exponential decay of the power spectrum of turbulence, J. Stat. Phys., 80 (1995), 579-602. doi: 10.1007/BF02178549.
[4]	A. Bronzi, C. Mondaini and R. Rosa, On the convergence of statistical solutions, work in progress.
[5]	M. Capinski and N. Cutland, A simple proof of existence of weak and statistical solutions of Navier-Stokes equations, Proc. Roy. Soc. London Ser. A, 436 (1992), 1-11. doi: 10.1098/rspa.1992.0001.
[6]	S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, Camassa-Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett., 81 (1998), 5338-5341. doi: 10.1103/PhysRevLett.81.5338.
[7]	S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in channels and pipes, Phys. Fluids, 11 (1999), 2343-2353. doi: 10.1063/1.870096.
[8]	S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, The Camassa-Holm equations and turbulence, Phys. D, 133 (1999), 49-65. doi: 10.1016/S0167-2789(99)00098-6.
[9]	P. Constantin and C. Foias, "Navier-Stokes Equations,'' Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.
[10]	P. Constantin, C. Foias and O. P. Manley, Effects of the forcing function on the energy spectrum in 2-D turbulence, Phys. Fluids, 6 (1994), 427-429. doi: 10.1063/1.868042.
[11]	P. Constantin and F. Ramos, Inviscid limit for damped and driven incompressible Navier-Stokes equations in $\mathbb{R}^2$, Comm. Math. Phys., 275 (2007), 529-551. doi: 10.1007/s00220-007-0310-7.
[12]	P. Constantin and J. Wu, Statistical solutions of the Navier-Stokes equations on the phase space of vorticity and the inviscid limits, J. Math. Phys., 38 (1997), 3031-3045. doi: 10.1063/1.532032.
[13]	F. Flandoli, An introduction to 3D stochastic fluid dynamics, in "SPDE in hydrodynamic: Recent progress and prospects,'' Lecture Notes in Math., 1942, Springer, Berlin, (2008), 51-150 doi: 10.1007/978-3-540-78493-7_2.
[14]	C. Foias, Statistical study of Navier-Stokes equations. I, Rend. Sem. Mat. Univ. Padova, 48 (1972), 219-348.
[15]	C. Foias, Statistical study of Navier-Stokes equations. II, Rend. Sem. Mat. Univ. Padova, 49 (1973), 9-123.
[16]	C. Foias, A functional approach to turbulence, Russian Math. Survey, 29 (1974), 293-326.
[17]	C. Foias, What do the Navier-Stokes equations tell us about turbulence?, in "Harmonic Analysis and Nonlinear Differential Equations'' (Riverside, CA, 1995), Contemp. Math., 208, Amer. Math. Soc., Providence, RI, (1997), 151-180. doi: 10.1090/conm/208/02739.
[18]	C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence. Advances in nonlinear mathematics and science, Phys. D, 152/153 (2001), 505-519. doi: 10.1016/S0167-2789(01)00191-9.
[19]	C. Foias, D. D. Holm and E. S. Titi, The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory, J. Dynam. Differential Equations, 14 (2002), 1-35. doi: 10.1023/A:1012984210582.
[20]	C. Foias, O. P. Manley, R. M. S. Rosa and R. Temam, Estimates for the energy cascade in three-dimensional turbulent flows, Comptes Rendus Acad. Sci. Paris, Série I, 333 (2001), 499-504. doi: 10.1016/S0764-4442(01)02008-0.
[21]	C. Foias, O. Manley, R. Rosa and R. Temam, "Navier-Stokes Equations and Turbulence,'' Encyclopedia of Mathematics and its Applications, Vol. 83, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511546754.
[22]	C. Foias and G. Prodi, Sur les solutions statistiques des équations de Navier-Stokes, Ann. Mat. Pura Appl. (4), 111 (1976), 307-330. doi: 10.1007/BF02411822.
[23]	C. Foias, R. M. S. Rosa and R. Temam, A note on statistical solutions of the three-dimensional Navier-Stokes equations: The time-dependent case, Comptes Rendus Acad. Sci. Paris, 348 (2010), 235-240. doi: 10.1016/j.crma.2009.12.017.
[24]	C. Foias, R. M. S. Rosa and R. Temam, A note on statistical solutions of the three-dimensional Navier-Stokes equations: The stationary case, Comptes Rendus Acad. Sci. Paris, 348 (2010), 347-353. doi: 10.1016/j.crma.2009.12.018.
[25]	C. Foias, R. Rosa and R. Temam, Properties of time-dependent statistical solutions of the three-dimensional Navier-Stokes equations, to appear in Annales de l'Institut Fourier, arXiv:1111.6257v1.
[26]	C. Foias, R. Rosa and R. Temam, Properties of stationary statistical solutions of the three-dimensional Navier-Stokes equations, in preparation.
[27]	A. V. Fursikov, The closure problem for the Friedman-Keller infinite chain of moment equations, corresponding to the Navier-Stokes system, in "Fundamental Problematic Issues in Turbulence" (Monte Verita, 1998), Trends Math., Birkhäuser, Basel, (1999), 17-24.
[28]	J. O. Hinze, "Turbulence: An Introduction to its Mechanism and Theory,'' Second edition, McGraw-Hill Book Co., Inc., New York, New York, 1975.
[29]	M. Holst, E. Lunasin and G. Tsogtgerel, Analysis of a general family of regularized Navier-Stokes and MHD models, J. Nonlinear Sci., 20 (2010), 523-567. doi: 10.1007/s00332-010-9066-x.
[30]	E. Hopf, Statistical hydromechanics and functional calculus, J. Rational Mech. Anal., 1 (1952), 87-123.
[31]	O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'' Revised English edition, Translated from the Russian by Richard A. Silverman, Gordon and Breach Science Publishers, New York-London, 1963.
[32]	A. S. Monin and A. M. Yaglom, "Statistical Fluid Mechanics: Mechanics of Turbulence,'' Volume 2, MIT Press, Cambridge, MA, 1975.
[33]	Yu. V. Prohorov, Convergence of random processes and limit theorems in probability theory, Teor. Veroyatnost. i Primenen., 1 (1956), 177-238.
[34]	F. Ramos, R. Rosa and R. Temam, Statistical estimates for channel flows driven by a pressure gradient, Phys. D, 237 (2008), 1368-1387. doi: 10.1016/j.physd.2008.03.013.
[35]	F. Ramos and E. S. Titi, Invariant measures for the 3D Navier-Stokes-Voigt equations and their Navier-Stokes limit, Discrete Contin. Dyn. Syst., 28 (2010), 375-403. doi: 10.3934/dcds.2010.28.375.
[36]	R. M. S. Rosa, Some results on the Navier-Stokes equations in connection with the statistical theory of stationary turbulence, Appl. Math., 47 (2002), 485-516. doi: 10.1023/A:1023297721804.
[37]	R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,'' Third edition, Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam-New York, (1984); Reedition in the AMS Chelsea series, AMS, Providence, 2001.
[38]	R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis,'' Second edition, CBMS-NSF Regional Conference Series in Applied Mathematics, 66, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. doi: 10.1137/1.9781611970050.
[39]	H. Tennekes and J. L. Lumley, "A First Course in Turbulence,'' MIT Press, Cambridge, Mass., 1972.
[40]	F. Topsøe, Compactness in spaces of measures, Studia Math., 36 (1970), 195-222.
[41]	F. Topsøe, "Topology and Measure,'' Lecture Notes in Mathematics, Vol. 133, Springer-Verlag, Berlin-New York, 1970.
[42]	F. Topsøe, Compactness and tightness in a space of measures with the topology of weak convergence, Math. Scand., 34 (1974), 187-210.
[43]	M. I. Višik and A. V. Foursikov, L'équation de Hopf, les solutions statistiques, les moments correspondant aux systèmes des équations paraboliques quasilinéaires, J. Math. Pures Appl. (9), 56 (1977), 85-122.
[44]	M. I. Višik and A. V. Fursikov, Translationally homogeneous statistical solutions and individual solutions with infinite energy of a system of Navier-Stokes equations, (Russian) Sibirsk. Mat. Zh., 19 (1978), 1005-1031, 1213.
[45]	M. I. Višik and A. Fursikov, "Mathematical Problems of Statistical Hydrodynamics,'' Kluwer, Dordrecht, 1988.
[46]	M. I. Višik, E. S. Titi and V. V. Chepyzhov, On the convergence of trajectory attractors of the three-dimensional Navier-Stokes $\alpha$-model as $\alpha \rightarrow 0$, Sb. Math., 198 (2007), 1703-1736. doi: 10.1070/SM2007v198n12ABEH003902.

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