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September  2022, 27(9): 5181-5203. doi: 10.3934/dcdsb.2021270

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 Published  September 2022 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 and Continuous Dynamical Systems - B, 2022, 27 (9) : 5181-5203. 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.

[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.

[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.

[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.

[5]

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

[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.

[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).

[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.

[9]

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

[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.

[11]

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

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

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

[22]

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

[23]

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

[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.

[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.

[26]

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

[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.
[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.

[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.

[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.

[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.

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

[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.

[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.

[36]

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

[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.

[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.

[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.

[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. 

[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.

[42]

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

[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.

[44]

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

[45]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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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.

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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.

[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.

[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.

[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.

[5]

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

[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.

[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).

[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.

[9]

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

[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.

[11]

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

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[21]

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

[22]

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

[23]

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

[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.

[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.

[26]

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

[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.
[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.

[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.

[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.

[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.

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

[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.

[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.

[36]

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

[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.

[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.

[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.

[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. 

[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.

[42]

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

[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.

[44]

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

[45]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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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.

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