doi: 10.3934/dcdsb.2021270
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On the zeroth law of turbulence for the stochastically forced Navier-Stokes equations

1. 

Department of Mathematics, University of California, Riverside, Riverside, CA 92507, USA

2. 

Department of Mathematics, Indiana University Bloomington, Bloomington, IN 47405, USA

*Corresponding author: Ali Pakzad

We dedicate this paper to the late Charlie Doering. He was an inspiring scientist and a wonderful person.

Received  April 2021 Revised  September 2021 Early access November 2021

We consider the three-dimensional stochastically forced Navier–Stokes equations subjected to white-in-time (colored-in-space) forcing in the absence of boundaries. Upper bounds of the mean value of the time-averaged energy dissipation rate are derived directly from the equations for weak (martingale) solutions. This estimate is consistent with the Kolmogorov dissipation law. Moreover, an additional hypothesis of energy balance implies the zeroth law of turbulence in the absence of a deterministic force.

Citation: Yat Tin Chow, Ali Pakzad. On the zeroth law of turbulence for the stochastically forced Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021270
References:
[1]

S. Albeverio, F. Flandoli and Y. Sinai, SPDE in Hydrodynamic: Recent Progress and Prospects, Lecture Notes in Mathematics, 1942. Springer-Verlag, Berlin; Fondazione C. I. M. E., Florence, 2008. doi: 10.1007/978-3-540-78493-7.  Google Scholar

[2]

A. Bensoussan and J. Frehse, Local solutions for stochastic Navier Stokes equations, M2AN Math. Model. Numer. Anal., 34 (2000), 241-273.  doi: 10.1051/m2an:2000140.  Google Scholar

[3]

A. BiswasM. S. JollyV. R. Martinez and E. S. Titi, Dissipation length scale estimates for turbulent flows: A wiener algebra approach, J. Nonlinear Sci., 24 (2014), 441-471.  doi: 10.1007/s00332-014-9195-8.  Google Scholar

[4]

A. Bensoussan and R. Temam, Equatios stochastique du type Navier-Stokes, J. Funct. Anal., 13 (1973), 195-222.  doi: 10.1016/0022-1236(73)90045-1.  Google Scholar

[5]

F. H. Busse, Bounds for turbulent shear flow, J. Fluid Mechanics, 41 (1970), 219-240.  doi: 10.1017/S0022112070000599.  Google Scholar

[6]

H. Breckner, Galerkin approximation and the strong solution of the Navier-Stokes equation, J. Appl. Math. Stochastic Anal., 13 (2000), 239-259.  doi: 10.1155/S1048953300000228.  Google Scholar

[7]

J. Bedrossian, A. Blumenthal and S. Punshon-Smith, The Batchelor spectrum of passive scalar turbulence in stochastic fluid mechanics, Communications on Pure and Applied Mathematics, 10.1002/cpa. 22022 (2021). Google Scholar

[8]

J. BedrossianM. Coti ZelatiS. Punshon-Smith and F. Weber, A sufficient condition for the Kolmogorov 4/5 law for stationary martingale solutions to the 3D Navier-Stokes equations, Comm. Math. Phys., 367 (2019), 1045-1075.  doi: 10.1007/s00220-019-03396-6.  Google Scholar

[9]

V. Barbu, Stabilization of Navier-Stokes Flows, Communications and Control Engineering Series. Springer, London, 2011. doi: 10.1007/978-0-85729-043-4.  Google Scholar

[10]

Z. Brzeźniak and S. Peszat, Infinite dimensional stochastic analysis, In Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet., R. Neth. Acad. Arts Sci., Amsterdam, 52 (2000), 85–98.  Google Scholar

[11]

A. J. Chorin, Numerical study of slightly visous flow, J. Fluid Mech., 57 (1973), 785-796.  doi: 10.1017/S0022112073002016.  Google Scholar

[12]

A. CheskidovP. Constantin and S. Friedlander, Energy conservation and Onsager's conjecture for the Euler equations, Nonlinearity, 21 (2008), 1233-1252.   Google Scholar

[13]

A. Cheskidov and X. Luo, Energy equality for the Navier-Stokes equations in weak-in-time Onsager spaces, Nonlinearity, 33 (2020), 1388-1403.  doi: 10.1088/1361-6544/ab60d3.  Google Scholar

[14]

M. Capiński and S. Peszat, Local existence and uniqueness of strong solutions to 3-D stochastic Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 185-200.  doi: 10.1007/PL00001415.  Google Scholar

[15]

P. Constantin and G. Iyer, A stochastic Lagrangian representation of the three-dimensional incompressible Navier-Stokes equations, Comm. Pure Appl. Math., 61 (2008), 330-345.  doi: 10.1002/cpa.20192.  Google Scholar

[16]

P. Constantin and G. Iyer, A Stochastic-Lagrangian approach to the Navier-Stokes equations in domains with boundary, Ann. Appl. Probab., 21 (2011), 1466-1492.  doi: 10.1214/10-AAP731.  Google Scholar

[17]

P. ConstantinE. Weinan and Edriss S. Titi, Onsager's conjecture on the energy conservation for solutions of Euler's equation, Commun. Math. Phys., 165 (1994), 207-209.  doi: 10.1007/BF02099744.  Google Scholar

[18]

T. CaraballoK. Liu and X. R. Mao, On stabilization of partial differential equations by noise, Nagoya Math. J., 161 (2001), 155-170.  doi: 10.1017/S0027763000022169.  Google Scholar

[19]

A. DebusscheN. Glatt-HoltzR. Temam and M. Ziane, Global existence and regularity for the 3D stochastic primitive equations of the ocean and atmosphere with multiplicative white noise, Nonlinearity, 25 (2012), 2093-2118.  doi: 10.1088/0951-7715/25/7/2093.  Google Scholar

[20]

C. R. Doering, The 3D navier-stokes problem, Annu. Rev. Fluid Mech., 41 (2009), 109-128.  doi: 10.1146/annurev.fluid.010908.165218.  Google Scholar

[21]

C. R. Doering and P. Constantin, Energy dissipation in shear driven turbulence, Physical Review Letters, 69 (1992), 1648.   Google Scholar

[22]

C. R. Doering and P. Constantin, Variational bounds on energy dissipation in incompressible flows. III. Convection, Phys. Rev. E, 53 (1996), 5957-5981.   Google Scholar

[23]

C. R. Doering and C. Foias, Energy dissipation in body-forced turbulence, J. Fluid Mech., 467 (2002), 289-306.  doi: 10.1017/S0022112002001386.  Google Scholar

[24]

J. Duchon andf R. Robert, Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations, Nonlinearity, 13 (2000), 249-255.  doi: 10.1088/0951-7715/13/1/312.  Google Scholar

[25]

V. DeCariaW. LaytonA. PakzadY. RongN. Sahin and H. Zhao, On the determination of the grad-div criterion, J. Math. Anal. Appl., 467 (2018), 1032-1037.  doi: 10.1016/j.jmaa.2018.07.040.  Google Scholar

[26]

L. C. Evans, An Introduction to Stochastic Differential Equations, American Mathematical Society, 2013. doi: 10.1090/mbk/082.  Google Scholar

[27] C. FoiasO. ManleyR. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and its Applications, 83. Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511546754.  Google Scholar
[28]

F. Flandoli and D. Gpolhk atarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.  Google Scholar

[29]

F. FlandoliM. GubinelliM. Hairer and M. Romito, Rigorous remarks about scaling laws in turbulent fluids, Comm. Math. Phys., 278 (2008), 1-29.  doi: 10.1007/s00220-007-0398-9.  Google Scholar

[30]

F. Flandoli and M. Romito, Partial regularity for the stochastic Navier-Stokes equations, Trans. Amer. Math. Soc., 354 (2002), 2207-2241.  doi: 10.1090/S0002-9947-02-02975-6.  Google Scholar

[31]

K. FellnerS. SonnerB. Q. Tang and D. D. Thuan, Stabilisation by noise on the boundary for a Chafee-Infante equation with dynamical boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4055-4078.  doi: 10.3934/dcdsb.2019050.  Google Scholar

[32] U. Frisch, Turbulence, The Legacy of A. N. Kolmogorov, Cambridge University Press, Cambridge, 1995.   Google Scholar
[33]

W. L. FanM. Jolly and A. Pakzad, Three-dimensional shear driven turbulence with noise at the boundary, Nonlinearity, 34 (2021), 4764-4786.  doi: 10.1088/1361-6544/abf84b.  Google Scholar

[34]

L. N. Howard, Bounds on flow quantities, Annual Review of Fluid Mechanics, 4 (1972), 473-494.  doi: 10.1146/annurev.fl.04.010172.002353.  Google Scholar

[35]

N. Glatt-Holtz, I. Kukavica, V. Vicol and M. Ziane, Existence and regularity of invariant measures for the three dimensional stochastic primitive equations, J. Math. Phys., 55 (2014), 051504, 34pp. doi: 10.1063/1.4875104.  Google Scholar

[36]

N. Glatt-Holtz and M. Ziane, Strong pathwise solutions of the stochastic Navier-Stokes system, Adv. Differential Equations, 14 (2009), 567-600.   Google Scholar

[37]

N. Jiang and W. J. Layton, Algorithms and models for turbulence not at statistical equilibrium, Comput. Math. Appl., 71 (2016), 2352-2372.  doi: 10.1016/j.camwa.2015.10.004.  Google Scholar

[38]

A. A. Kwiecinska, Stabilization of partial differential equations by noise, Stochastic Process. Appl., 79 (1999), 179-184.  doi: 10.1016/S0304-4149(98)00080-5.  Google Scholar

[39]

A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Translated from the Russian by V. Levin; Turbulence and stochastic processes: Kolmogorov's ideas 50 years on, Proc. Roy. Soc. London Ser. A, 434 (1991), 9-13.  doi: 10.1098/rspa.1991.0075.  Google Scholar

[40]

I. KukavicaK. Uğurlu and M. Ziane, On the Galerkin approximation and strong norm bounds for the stochastic Navier-Stokes equations with multiplicative noise, Differential Integral Equations, 31 (2018), 173-186.   Google Scholar

[41]

J. U. Kim, Strong solutions of the stochastic Navier-Stokes equations in $\Bbb R^3$, Indiana Univ. Math. J., 59 (2010), 1417-1450.  doi: 10.1512/iumj.2010.59.3930.  Google Scholar

[42]

K. Kean, W. Layton and M. Schneier, Clipping over dissipation in turbulence models, preprint, arXiv: 2109.12107. Google Scholar

[43]

R. R. Kerswell, Variational bounds on shear-driven turbulence and turbulent Boussinesq convection, Physica D, 100 (1997), 355-376.  doi: 10.1016/S0167-2789(96)00227-8.  Google Scholar

[44]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta. Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[45]

J. L. Lions, Quelques Méthodes De Résolution Des Problemes Aux Limites Non Linéaires, Dunod; Gauthier-Villars, Paris 1969.  Google Scholar

[46]

T. M. Leslie and R. Shvydkoy, Conditions implying energy equality for weak solutions of the Navier-Stokes equations, SIAM J. Math. Anal., 50 (2018), 870-890.  doi: 10.1137/16M1104147.  Google Scholar

[47]

W. J. Layton, Energy dissipation in the Smagorinsky model of turbulence, Appl. Math. Lett., 59 (2016), 56-59.  doi: 10.1016/j.aml.2016.03.008.  Google Scholar

[48]

W. J. Layton, Energy dissipation bounds for shear flows for a model in large eddy simulation, Math. Comput. Modelling, 35 (2002), 1445-1451.  doi: 10.1016/S0895-7177(02)00095-X.  Google Scholar

[49]

C. Marchioro, Remark on the energy dissipation in shear driven turbulence, Phys. D, 74 (1994), 395-398.  doi: 10.1016/0167-2789(94)90203-8.  Google Scholar

[50]

R. Mikulevicius and B. L. Rozovskii, Stochastic Navier-Stokes equations for turbulent flows, SIAM J. Math. Anal., 35 (2004), 1250-1310.  doi: 10.1137/S0036141002409167.  Google Scholar

[51]

R. Mikulevicius and B. L. Rozovskii, Global $L^2$ -solutions of stochastic Navier-Stokes equations, Ann. Probab., 33 (2005), 137-176.  doi: 10.1214/009117904000000630.  Google Scholar

[52]

W. S. Ożański and B. C. Pooley, Leray's fundamental work on the Navier-Stokes equations: A modern review of "Sur le mouvement d'un liquide visqueux emplissant l'espace", preprint, arXiv: 1708.09787. Google Scholar

[53]

A. Pakzad, Damping functions correct over-dissipation of the Smagorinsky model, Math. Methods Appl. Sci., 40 (2017), 5933-5945.  doi: 10.1002/mma.4444.  Google Scholar

[54]

A. Pakzad, Analysis of mesh effects on turbulence statistics, J. Math. Anal. Appl., 475 (2019), 839-860.  doi: 10.1016/j.jmaa.2019.02.075.  Google Scholar

[55]

A. Pakzad, On the long time behavior of time relaxation model of fluids, Phys. D, 408 (2020), 132509.  doi: 10.1016/j.physd.2020.132509.  Google Scholar

[56] S. B. Pope, Turbulent Flows, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511840531.  Google Scholar
[57]

M. Romito, Existence of martingale and stationary suitable weak solutions for a stochastic Navier–Stokes system, Stochastics, 82 (2010), 327-337.  doi: 10.1080/17442501003721542.  Google Scholar

[58]

K. R. Sreenivasan, An update on the energy dissipation rate in isotropic turbulence, Phys. Fluids, 10 (1998), 528-529.  doi: 10.1063/1.869575.  Google Scholar

[59]

V. Scheffer, Hausdorff measure and the Navier-Stokes equations, Comm. Math. Phys., 55 (1977), 97-112.  doi: 10.1007/BF01626512.  Google Scholar

[60]

J. C. Vassilicos, Dissipation in turbulent flows, Annu. Rev. Fluid Mech., 47 (2015), 95–114.  Google Scholar

[61]

M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, 9 Springer Science & Business Media, 2012. Google Scholar

[62]

D. Wang and H. Wang, Global existence of martingale solutions to the three-dimensional stochastic compressible Navier-Stokes equations, Differential Integral Equations, 28 (2015), 1105-1154.   Google Scholar

[63]

X. Wang, Approximation of stationary statistical properties of dissipative dynamical systems: Time discretization, Math. Compu., 79 (2010), 259-280.  doi: 10.1090/S0025-5718-09-02256-X.  Google Scholar

[64]

X. Wang, Effect of tangential derivative in the boundary layer on time averaged energy dissipation rate, Physica D: Nonlinear Phenomena, 144 (2000), 142-153.  doi: 10.1016/S0167-2789(00)00066-X.  Google Scholar

[65]

X. Wang, Time-averaged energy dissipation rate for shear driven flows in ${\bf R}^n$, Phys. D, 99 (1997), 555-563.  doi: 10.1016/S0167-2789(96)00161-3.  Google Scholar

show all references

References:
[1]

S. Albeverio, F. Flandoli and Y. Sinai, SPDE in Hydrodynamic: Recent Progress and Prospects, Lecture Notes in Mathematics, 1942. Springer-Verlag, Berlin; Fondazione C. I. M. E., Florence, 2008. doi: 10.1007/978-3-540-78493-7.  Google Scholar

[2]

A. Bensoussan and J. Frehse, Local solutions for stochastic Navier Stokes equations, M2AN Math. Model. Numer. Anal., 34 (2000), 241-273.  doi: 10.1051/m2an:2000140.  Google Scholar

[3]

A. BiswasM. S. JollyV. R. Martinez and E. S. Titi, Dissipation length scale estimates for turbulent flows: A wiener algebra approach, J. Nonlinear Sci., 24 (2014), 441-471.  doi: 10.1007/s00332-014-9195-8.  Google Scholar

[4]

A. Bensoussan and R. Temam, Equatios stochastique du type Navier-Stokes, J. Funct. Anal., 13 (1973), 195-222.  doi: 10.1016/0022-1236(73)90045-1.  Google Scholar

[5]

F. H. Busse, Bounds for turbulent shear flow, J. Fluid Mechanics, 41 (1970), 219-240.  doi: 10.1017/S0022112070000599.  Google Scholar

[6]

H. Breckner, Galerkin approximation and the strong solution of the Navier-Stokes equation, J. Appl. Math. Stochastic Anal., 13 (2000), 239-259.  doi: 10.1155/S1048953300000228.  Google Scholar

[7]

J. Bedrossian, A. Blumenthal and S. Punshon-Smith, The Batchelor spectrum of passive scalar turbulence in stochastic fluid mechanics, Communications on Pure and Applied Mathematics, 10.1002/cpa. 22022 (2021). Google Scholar

[8]

J. BedrossianM. Coti ZelatiS. Punshon-Smith and F. Weber, A sufficient condition for the Kolmogorov 4/5 law for stationary martingale solutions to the 3D Navier-Stokes equations, Comm. Math. Phys., 367 (2019), 1045-1075.  doi: 10.1007/s00220-019-03396-6.  Google Scholar

[9]

V. Barbu, Stabilization of Navier-Stokes Flows, Communications and Control Engineering Series. Springer, London, 2011. doi: 10.1007/978-0-85729-043-4.  Google Scholar

[10]

Z. Brzeźniak and S. Peszat, Infinite dimensional stochastic analysis, In Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet., R. Neth. Acad. Arts Sci., Amsterdam, 52 (2000), 85–98.  Google Scholar

[11]

A. J. Chorin, Numerical study of slightly visous flow, J. Fluid Mech., 57 (1973), 785-796.  doi: 10.1017/S0022112073002016.  Google Scholar

[12]

A. CheskidovP. Constantin and S. Friedlander, Energy conservation and Onsager's conjecture for the Euler equations, Nonlinearity, 21 (2008), 1233-1252.   Google Scholar

[13]

A. Cheskidov and X. Luo, Energy equality for the Navier-Stokes equations in weak-in-time Onsager spaces, Nonlinearity, 33 (2020), 1388-1403.  doi: 10.1088/1361-6544/ab60d3.  Google Scholar

[14]

M. Capiński and S. Peszat, Local existence and uniqueness of strong solutions to 3-D stochastic Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 185-200.  doi: 10.1007/PL00001415.  Google Scholar

[15]

P. Constantin and G. Iyer, A stochastic Lagrangian representation of the three-dimensional incompressible Navier-Stokes equations, Comm. Pure Appl. Math., 61 (2008), 330-345.  doi: 10.1002/cpa.20192.  Google Scholar

[16]

P. Constantin and G. Iyer, A Stochastic-Lagrangian approach to the Navier-Stokes equations in domains with boundary, Ann. Appl. Probab., 21 (2011), 1466-1492.  doi: 10.1214/10-AAP731.  Google Scholar

[17]

P. ConstantinE. Weinan and Edriss S. Titi, Onsager's conjecture on the energy conservation for solutions of Euler's equation, Commun. Math. Phys., 165 (1994), 207-209.  doi: 10.1007/BF02099744.  Google Scholar

[18]

T. CaraballoK. Liu and X. R. Mao, On stabilization of partial differential equations by noise, Nagoya Math. J., 161 (2001), 155-170.  doi: 10.1017/S0027763000022169.  Google Scholar

[19]

A. DebusscheN. Glatt-HoltzR. Temam and M. Ziane, Global existence and regularity for the 3D stochastic primitive equations of the ocean and atmosphere with multiplicative white noise, Nonlinearity, 25 (2012), 2093-2118.  doi: 10.1088/0951-7715/25/7/2093.  Google Scholar

[20]

C. R. Doering, The 3D navier-stokes problem, Annu. Rev. Fluid Mech., 41 (2009), 109-128.  doi: 10.1146/annurev.fluid.010908.165218.  Google Scholar

[21]

C. R. Doering and P. Constantin, Energy dissipation in shear driven turbulence, Physical Review Letters, 69 (1992), 1648.   Google Scholar

[22]

C. R. Doering and P. Constantin, Variational bounds on energy dissipation in incompressible flows. III. Convection, Phys. Rev. E, 53 (1996), 5957-5981.   Google Scholar

[23]

C. R. Doering and C. Foias, Energy dissipation in body-forced turbulence, J. Fluid Mech., 467 (2002), 289-306.  doi: 10.1017/S0022112002001386.  Google Scholar

[24]

J. Duchon andf R. Robert, Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations, Nonlinearity, 13 (2000), 249-255.  doi: 10.1088/0951-7715/13/1/312.  Google Scholar

[25]

V. DeCariaW. LaytonA. PakzadY. RongN. Sahin and H. Zhao, On the determination of the grad-div criterion, J. Math. Anal. Appl., 467 (2018), 1032-1037.  doi: 10.1016/j.jmaa.2018.07.040.  Google Scholar

[26]

L. C. Evans, An Introduction to Stochastic Differential Equations, American Mathematical Society, 2013. doi: 10.1090/mbk/082.  Google Scholar

[27] C. FoiasO. ManleyR. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and its Applications, 83. Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511546754.  Google Scholar
[28]

F. Flandoli and D. Gpolhk atarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.  Google Scholar

[29]

F. FlandoliM. GubinelliM. Hairer and M. Romito, Rigorous remarks about scaling laws in turbulent fluids, Comm. Math. Phys., 278 (2008), 1-29.  doi: 10.1007/s00220-007-0398-9.  Google Scholar

[30]

F. Flandoli and M. Romito, Partial regularity for the stochastic Navier-Stokes equations, Trans. Amer. Math. Soc., 354 (2002), 2207-2241.  doi: 10.1090/S0002-9947-02-02975-6.  Google Scholar

[31]

K. FellnerS. SonnerB. Q. Tang and D. D. Thuan, Stabilisation by noise on the boundary for a Chafee-Infante equation with dynamical boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4055-4078.  doi: 10.3934/dcdsb.2019050.  Google Scholar

[32] U. Frisch, Turbulence, The Legacy of A. N. Kolmogorov, Cambridge University Press, Cambridge, 1995.   Google Scholar
[33]

W. L. FanM. Jolly and A. Pakzad, Three-dimensional shear driven turbulence with noise at the boundary, Nonlinearity, 34 (2021), 4764-4786.  doi: 10.1088/1361-6544/abf84b.  Google Scholar

[34]

L. N. Howard, Bounds on flow quantities, Annual Review of Fluid Mechanics, 4 (1972), 473-494.  doi: 10.1146/annurev.fl.04.010172.002353.  Google Scholar

[35]

N. Glatt-Holtz, I. Kukavica, V. Vicol and M. Ziane, Existence and regularity of invariant measures for the three dimensional stochastic primitive equations, J. Math. Phys., 55 (2014), 051504, 34pp. doi: 10.1063/1.4875104.  Google Scholar

[36]

N. Glatt-Holtz and M. Ziane, Strong pathwise solutions of the stochastic Navier-Stokes system, Adv. Differential Equations, 14 (2009), 567-600.   Google Scholar

[37]

N. Jiang and W. J. Layton, Algorithms and models for turbulence not at statistical equilibrium, Comput. Math. Appl., 71 (2016), 2352-2372.  doi: 10.1016/j.camwa.2015.10.004.  Google Scholar

[38]

A. A. Kwiecinska, Stabilization of partial differential equations by noise, Stochastic Process. Appl., 79 (1999), 179-184.  doi: 10.1016/S0304-4149(98)00080-5.  Google Scholar

[39]

A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Translated from the Russian by V. Levin; Turbulence and stochastic processes: Kolmogorov's ideas 50 years on, Proc. Roy. Soc. London Ser. A, 434 (1991), 9-13.  doi: 10.1098/rspa.1991.0075.  Google Scholar

[40]

I. KukavicaK. Uğurlu and M. Ziane, On the Galerkin approximation and strong norm bounds for the stochastic Navier-Stokes equations with multiplicative noise, Differential Integral Equations, 31 (2018), 173-186.   Google Scholar

[41]

J. U. Kim, Strong solutions of the stochastic Navier-Stokes equations in $\Bbb R^3$, Indiana Univ. Math. J., 59 (2010), 1417-1450.  doi: 10.1512/iumj.2010.59.3930.  Google Scholar

[42]

K. Kean, W. Layton and M. Schneier, Clipping over dissipation in turbulence models, preprint, arXiv: 2109.12107. Google Scholar

[43]

R. R. Kerswell, Variational bounds on shear-driven turbulence and turbulent Boussinesq convection, Physica D, 100 (1997), 355-376.  doi: 10.1016/S0167-2789(96)00227-8.  Google Scholar

[44]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta. Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[45]

J. L. Lions, Quelques Méthodes De Résolution Des Problemes Aux Limites Non Linéaires, Dunod; Gauthier-Villars, Paris 1969.  Google Scholar

[46]

T. M. Leslie and R. Shvydkoy, Conditions implying energy equality for weak solutions of the Navier-Stokes equations, SIAM J. Math. Anal., 50 (2018), 870-890.  doi: 10.1137/16M1104147.  Google Scholar

[47]

W. J. Layton, Energy dissipation in the Smagorinsky model of turbulence, Appl. Math. Lett., 59 (2016), 56-59.  doi: 10.1016/j.aml.2016.03.008.  Google Scholar

[48]

W. J. Layton, Energy dissipation bounds for shear flows for a model in large eddy simulation, Math. Comput. Modelling, 35 (2002), 1445-1451.  doi: 10.1016/S0895-7177(02)00095-X.  Google Scholar

[49]

C. Marchioro, Remark on the energy dissipation in shear driven turbulence, Phys. D, 74 (1994), 395-398.  doi: 10.1016/0167-2789(94)90203-8.  Google Scholar

[50]

R. Mikulevicius and B. L. Rozovskii, Stochastic Navier-Stokes equations for turbulent flows, SIAM J. Math. Anal., 35 (2004), 1250-1310.  doi: 10.1137/S0036141002409167.  Google Scholar

[51]

R. Mikulevicius and B. L. Rozovskii, Global $L^2$ -solutions of stochastic Navier-Stokes equations, Ann. Probab., 33 (2005), 137-176.  doi: 10.1214/009117904000000630.  Google Scholar

[52]

W. S. Ożański and B. C. Pooley, Leray's fundamental work on the Navier-Stokes equations: A modern review of "Sur le mouvement d'un liquide visqueux emplissant l'espace", preprint, arXiv: 1708.09787. Google Scholar

[53]

A. Pakzad, Damping functions correct over-dissipation of the Smagorinsky model, Math. Methods Appl. Sci., 40 (2017), 5933-5945.  doi: 10.1002/mma.4444.  Google Scholar

[54]

A. Pakzad, Analysis of mesh effects on turbulence statistics, J. Math. Anal. Appl., 475 (2019), 839-860.  doi: 10.1016/j.jmaa.2019.02.075.  Google Scholar

[55]

A. Pakzad, On the long time behavior of time relaxation model of fluids, Phys. D, 408 (2020), 132509.  doi: 10.1016/j.physd.2020.132509.  Google Scholar

[56] S. B. Pope, Turbulent Flows, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511840531.  Google Scholar
[57]

M. Romito, Existence of martingale and stationary suitable weak solutions for a stochastic Navier–Stokes system, Stochastics, 82 (2010), 327-337.  doi: 10.1080/17442501003721542.  Google Scholar

[58]

K. R. Sreenivasan, An update on the energy dissipation rate in isotropic turbulence, Phys. Fluids, 10 (1998), 528-529.  doi: 10.1063/1.869575.  Google Scholar

[59]

V. Scheffer, Hausdorff measure and the Navier-Stokes equations, Comm. Math. Phys., 55 (1977), 97-112.  doi: 10.1007/BF01626512.  Google Scholar

[60]

J. C. Vassilicos, Dissipation in turbulent flows, Annu. Rev. Fluid Mech., 47 (2015), 95–114.  Google Scholar

[61]

M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, 9 Springer Science & Business Media, 2012. Google Scholar

[62]

D. Wang and H. Wang, Global existence of martingale solutions to the three-dimensional stochastic compressible Navier-Stokes equations, Differential Integral Equations, 28 (2015), 1105-1154.   Google Scholar

[63]

X. Wang, Approximation of stationary statistical properties of dissipative dynamical systems: Time discretization, Math. Compu., 79 (2010), 259-280.  doi: 10.1090/S0025-5718-09-02256-X.  Google Scholar

[64]

X. Wang, Effect of tangential derivative in the boundary layer on time averaged energy dissipation rate, Physica D: Nonlinear Phenomena, 144 (2000), 142-153.  doi: 10.1016/S0167-2789(00)00066-X.  Google Scholar

[65]

X. Wang, Time-averaged energy dissipation rate for shear driven flows in ${\bf R}^n$, Phys. D, 99 (1997), 555-563.  doi: 10.1016/S0167-2789(96)00161-3.  Google Scholar

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