# American Institute of Mathematical Sciences

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

 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

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.
Citation: Anne Bronzi, Ricardo Rosa. On the convergence of statistical solutions of the 3D Navier-Stokes-$\alpha$ model as $\alpha$ vanishes. Discrete & Continuous Dynamical Systems - A, 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, (2006). Google Scholar [2] G. K. Batchelor, "The Theory of Homogeneous Turbulence,'', Cambridge Monographs on Mechanics and Applied Mathematics, (1953). Google Scholar [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. doi: 10.1007/BF02178549. Google Scholar [4] A. Bronzi, C. Mondaini and R. Rosa, On the convergence of statistical solutions,, work in progress., (). Google Scholar [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. doi: 10.1098/rspa.1992.0001. Google Scholar [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. doi: 10.1103/PhysRevLett.81.5338. Google Scholar [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. doi: 10.1063/1.870096. Google Scholar [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. doi: 10.1016/S0167-2789(99)00098-6. Google Scholar [9] P. Constantin and C. Foias, "Navier-Stokes Equations,'', Chicago Lectures in Mathematics, (1988). Google Scholar [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. doi: 10.1063/1.868042. Google Scholar [11] P. Constantin and F. Ramos, Inviscid limit for damped and driven incompressible Navier-Stokes equations in $\mathbbR^2$,, Comm. Math. Phys., 275 (2007), 529. doi: 10.1007/s00220-007-0310-7. Google Scholar [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. doi: 10.1063/1.532032. Google Scholar [13] F. Flandoli, An introduction to 3D stochastic fluid dynamics,, in, 1942 (2008), 51. doi: 10.1007/978-3-540-78493-7_2. Google Scholar [14] C. Foias, Statistical study of Navier-Stokes equations. I,, Rend. Sem. Mat. Univ. Padova, 48 (1972), 219. Google Scholar [15] C. Foias, Statistical study of Navier-Stokes equations. II,, Rend. Sem. Mat. Univ. Padova, 49 (1973), 9. Google Scholar [16] C. Foias, A functional approach to turbulence,, Russian Math. Survey, 29 (1974), 293. Google Scholar [17] C. Foias, What do the Navier-Stokes equations tell us about turbulence?,, in, 208 (1997), 151. doi: 10.1090/conm/208/02739. Google Scholar [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. doi: 10.1016/S0167-2789(01)00191-9. Google Scholar [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. doi: 10.1023/A:1012984210582. Google Scholar [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, 333 (2001), 499. doi: 10.1016/S0764-4442(01)02008-0. Google Scholar [21] C. Foias, O. Manley, R. Rosa and R. Temam, "Navier-Stokes Equations and Turbulence,'', Encyclopedia of Mathematics and its Applications, (2001). doi: 10.1017/CBO9780511546754. Google Scholar [22] C. Foias and G. Prodi, Sur les solutions statistiques des équations de Navier-Stokes,, Ann. Mat. Pura Appl. (4), 111 (1976), 307. doi: 10.1007/BF02411822. Google Scholar [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. doi: 10.1016/j.crma.2009.12.017. Google Scholar [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. doi: 10.1016/j.crma.2009.12.018. Google Scholar [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, (). Google Scholar [26] C. Foias, R. Rosa and R. Temam, Properties of stationary statistical solutions of the three-dimensional Navier-Stokes equations,, in preparation., (). Google Scholar [27] A. V. Fursikov, The closure problem for the Friedman-Keller infinite chain of moment equations, corresponding to the Navier-Stokes system,, in, (1999), 17. Google Scholar [28] J. O. Hinze, "Turbulence: An Introduction to its Mechanism and Theory,'', Second edition, (1975). Google Scholar [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. doi: 10.1007/s00332-010-9066-x. Google Scholar [30] E. Hopf, Statistical hydromechanics and functional calculus,, J. Rational Mech. Anal., 1 (1952), 87. Google Scholar [31] O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'', Revised English edition, (1963). Google Scholar [32] A. S. Monin and A. M. Yaglom, "Statistical Fluid Mechanics: Mechanics of Turbulence,'', Volume 2, (1975). Google Scholar [33] Yu. V. Prohorov, Convergence of random processes and limit theorems in probability theory,, Teor. Veroyatnost. i Primenen., 1 (1956), 177. Google Scholar [34] F. Ramos, R. Rosa and R. Temam, Statistical estimates for channel flows driven by a pressure gradient,, Phys. D, 237 (2008), 1368. doi: 10.1016/j.physd.2008.03.013. Google Scholar [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. doi: 10.3934/dcds.2010.28.375. Google Scholar [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. doi: 10.1023/A:1023297721804. Google Scholar [37] R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,'', Third edition, (1984). Google Scholar [38] R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis,'', Second edition, 66 (1995). doi: 10.1137/1.9781611970050. Google Scholar [39] H. Tennekes and J. L. Lumley, "A First Course in Turbulence,'', MIT Press, (1972). Google Scholar [40] F. Topsøe, Compactness in spaces of measures,, Studia Math., 36 (1970), 195. Google Scholar [41] F. Topsøe, "Topology and Measure,'', Lecture Notes in Mathematics, (1970). Google Scholar [42] F. Topsøe, Compactness and tightness in a space of measures with the topology of weak convergence,, Math. Scand., 34 (1974), 187. Google Scholar [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. Google Scholar [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. Google Scholar [45] M. I. Višik and A. Fursikov, "Mathematical Problems of Statistical Hydrodynamics,'', Kluwer, (1988). Google Scholar [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. doi: 10.1070/SM2007v198n12ABEH003902. Google Scholar

show all references

##### References:
 [1] C. D. Aliprantis and K. C. Border, "Infinite Dimensional Analysis. A Hitchhiker's Guide,'', Third edition, (2006). Google Scholar [2] G. K. Batchelor, "The Theory of Homogeneous Turbulence,'', Cambridge Monographs on Mechanics and Applied Mathematics, (1953). Google Scholar [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. doi: 10.1007/BF02178549. Google Scholar [4] A. Bronzi, C. Mondaini and R. Rosa, On the convergence of statistical solutions,, work in progress., (). Google Scholar [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. doi: 10.1098/rspa.1992.0001. Google Scholar [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. doi: 10.1103/PhysRevLett.81.5338. Google Scholar [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. doi: 10.1063/1.870096. Google Scholar [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. doi: 10.1016/S0167-2789(99)00098-6. Google Scholar [9] P. Constantin and C. Foias, "Navier-Stokes Equations,'', Chicago Lectures in Mathematics, (1988). Google Scholar [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. doi: 10.1063/1.868042. Google Scholar [11] P. Constantin and F. Ramos, Inviscid limit for damped and driven incompressible Navier-Stokes equations in $\mathbbR^2$,, Comm. Math. Phys., 275 (2007), 529. doi: 10.1007/s00220-007-0310-7. Google Scholar [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. doi: 10.1063/1.532032. Google Scholar [13] F. Flandoli, An introduction to 3D stochastic fluid dynamics,, in, 1942 (2008), 51. doi: 10.1007/978-3-540-78493-7_2. Google Scholar [14] C. Foias, Statistical study of Navier-Stokes equations. I,, Rend. Sem. Mat. Univ. Padova, 48 (1972), 219. Google Scholar [15] C. Foias, Statistical study of Navier-Stokes equations. II,, Rend. Sem. Mat. Univ. Padova, 49 (1973), 9. Google Scholar [16] C. Foias, A functional approach to turbulence,, Russian Math. Survey, 29 (1974), 293. Google Scholar [17] C. Foias, What do the Navier-Stokes equations tell us about turbulence?,, in, 208 (1997), 151. doi: 10.1090/conm/208/02739. Google Scholar [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. doi: 10.1016/S0167-2789(01)00191-9. Google Scholar [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. doi: 10.1023/A:1012984210582. Google Scholar [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, 333 (2001), 499. doi: 10.1016/S0764-4442(01)02008-0. Google Scholar [21] C. Foias, O. Manley, R. Rosa and R. Temam, "Navier-Stokes Equations and Turbulence,'', Encyclopedia of Mathematics and its Applications, (2001). doi: 10.1017/CBO9780511546754. Google Scholar [22] C. Foias and G. Prodi, Sur les solutions statistiques des équations de Navier-Stokes,, Ann. Mat. Pura Appl. (4), 111 (1976), 307. doi: 10.1007/BF02411822. Google Scholar [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. doi: 10.1016/j.crma.2009.12.017. Google Scholar [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. doi: 10.1016/j.crma.2009.12.018. Google Scholar [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, (). Google Scholar [26] C. Foias, R. Rosa and R. Temam, Properties of stationary statistical solutions of the three-dimensional Navier-Stokes equations,, in preparation., (). Google Scholar [27] A. V. Fursikov, The closure problem for the Friedman-Keller infinite chain of moment equations, corresponding to the Navier-Stokes system,, in, (1999), 17. Google Scholar [28] J. O. Hinze, "Turbulence: An Introduction to its Mechanism and Theory,'', Second edition, (1975). Google Scholar [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. doi: 10.1007/s00332-010-9066-x. Google Scholar [30] E. Hopf, Statistical hydromechanics and functional calculus,, J. Rational Mech. Anal., 1 (1952), 87. Google Scholar [31] O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'', Revised English edition, (1963). Google Scholar [32] A. S. Monin and A. M. Yaglom, "Statistical Fluid Mechanics: Mechanics of Turbulence,'', Volume 2, (1975). Google Scholar [33] Yu. V. Prohorov, Convergence of random processes and limit theorems in probability theory,, Teor. Veroyatnost. i Primenen., 1 (1956), 177. Google Scholar [34] F. Ramos, R. Rosa and R. Temam, Statistical estimates for channel flows driven by a pressure gradient,, Phys. D, 237 (2008), 1368. doi: 10.1016/j.physd.2008.03.013. Google Scholar [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. doi: 10.3934/dcds.2010.28.375. Google Scholar [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. doi: 10.1023/A:1023297721804. Google Scholar [37] R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,'', Third edition, (1984). Google Scholar [38] R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis,'', Second edition, 66 (1995). doi: 10.1137/1.9781611970050. Google Scholar [39] H. Tennekes and J. L. Lumley, "A First Course in Turbulence,'', MIT Press, (1972). Google Scholar [40] F. Topsøe, Compactness in spaces of measures,, Studia Math., 36 (1970), 195. Google Scholar [41] F. Topsøe, "Topology and Measure,'', Lecture Notes in Mathematics, (1970). Google Scholar [42] F. Topsøe, Compactness and tightness in a space of measures with the topology of weak convergence,, Math. Scand., 34 (1974), 187. Google Scholar [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. Google Scholar [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. Google Scholar [45] M. I. Višik and A. Fursikov, "Mathematical Problems of Statistical Hydrodynamics,'', Kluwer, (1988). Google Scholar [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. doi: 10.1070/SM2007v198n12ABEH003902. Google Scholar
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