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November  2018, 17(6): 2593-2621. doi: 10.3934/cpaa.2018123

The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior

G. Deugoué 1, and  T. Tachim Medjo 2,,

1. 

Department of Mathematics and Computer Science, University of Dschang, P.O. BOX 67, Dschang, Cameroon
2. 

Department of Mathematics, Florida International University, MMC, Miami, Florida 33199, USA

* Corresponding author

Received  October 2016 Revised  July 2017 Published  June 2018

Keywords: Globally modfied, Navier-Stokes, stochastic, strong solutions, stability.
Mathematics Subject Classification: 35R60, 35Q35, 60H15, 76M35, 86A05.
Citation: G. Deugoué, T. Tachim Medjo. The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2593-2621. doi: 10.3934/cpaa.2018123
References:
[1]

D. Barbato, M. Barsati, H. Bessaih and F. Flandoli, Some rigorous results on a stochastic Goy model, J. Stat. Phys., 125 (2006), 677-716.  doi: 10.1007/s10955-006-9203-y.

[2]

A. Bensoussan, Stochastic Navier-Stokes equations, Acta Appl. Math., 38 (1995), 267-304.  doi: 10.1007/BF00996149.

[3]

A. Bensoussan and R. Temam, Equations stochastiques de type Navier-Stokes, J. Funct. Anal., 13 (1973), 195-222. 

[4]

H. Bessaih and A. Millet, Large deviation principle and inviscid shell models, Electron. J. Probab., 14 (2009), 2551-2579.  doi: 10.1214/EJP.v14-719.

[5]

H. Bessaih, F. Flandoli and E. S. Titi, Stochastic attractors for shell phenomenological models of turbulence, J. Stat. Phys., 140 (2010), 688-717.  doi: 10.1007/s10955-010-0010-0.

[6]

T. Caraballo, P. E. Kloeden and J. Real, Unique strong solutions and V-attractors of a 3- dimensional system of globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 6 (2006), 411-436.  doi: 10.1515/ans-2006-0304.

[7]

T. Caraballo and P. E. Kloeden, The three-dimensional globally modified Navier-Stokes equations: recent developments, Recent trends in Dynamical Systems, Springer Proc. Math. Stat., 35, 473-492 Springer, Basel, 2013. doi: 10.1007/978-3-0348-0451-6_18.

[8]

I. Chueshov and A. Millet, Stochastic 2D hydrodynamical type systems: well posedness and large deviations, Appl. Math. Optim., 61 (2010), 379-420.  doi: 10.1007/s00245-009-9091-z.

[9]

P. Constantin, Near Identity Transformations for the Navier-Stokes Equations, in Handbook of Mathematical Fluid Dynamics, Vol. Ⅱ, 117-141, North-Holland, Amsterdam, 2003. doi: 10.1016/S1874-5792(03)80006-X.

[10]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its applications, vol. 44, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.

[11]

A. Debussche, N. Glatt-Holtz and R. Temam, Local martingale and pathwise solutions for an abstract fluids model, Phys. D, 240 (2011), 1123-1144.  doi: 10.1016/j.physd.2011.03.009.

[12]

G. Deugoué and J. K. Djoko, On the time discretization for the globally modified 3- dimensional Navier-Stokes equations, J. Comput. Appl. Math., 235 (2011), 2015-2029.  doi: 10.1016/j.cam.2010.10.003.

[13]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 307-391.  doi: 10.1007/BF01192467.

[14]

F. Flandoli, An introduction to 3d stochastic fluid dynamics, SPDE in Hydrodynamic: Recent Progress and Prospects, Lecture Notes in Mathematics, vol. 1992, 51-150, Springer Berlin, Heidelberg, 2008. doi: 10. 1007/978-3-540-78493-7_2.

[15]

F. Flandoli and B. Maslowski, Ergodicity of the 2D Navier-Stokes equation under random perturbations, Commun. Math. Phys., 172 (1995), 119-141. 

[16]

F. Flandoli, M. Gubinelli, M. Hairer and M. Romito, Rigorous remarks about scaling laws in turbulent fluid, Commun. Math. Phys., 278 (2008), 1-29.  doi: 10.1007/s00220-007-0398-9.

[17] I. I. Gikhman and A. V. Skorohod, Stochastic Differential Equations, Springer-Verlag, Berlin, 1972. 
[18]

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

[19]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Second edition, North-Holland, Kodansha, 1989.

[20]

A. Jakubowski, The almost sure Skorokhod representation for subsequences in nonmetric spaces, translated from Teor. Veroyatnost. i Primenen, 42 (1997), 209-216.  doi: 10.1137/S0040585X97976052.

[21]

P. E. Kloeden and J. Valero, The weak connectedness of the attainability set of weak solutions of the three-dimensional Navier-Stokes equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci, 463 (2007), 1491-1508.  doi: 10.1098/rspa.2007.1831.

[22]

P. E. Kloeden, J. A. Langa and J. Real, Pullback V-attractors of the three dimensional globally modified Navier-Stokes equations: existence and finite fractal dimension, Commun. Pure Appl. Anal., 6 (2007), 937-955.  doi: 10.3934/cpaa.2007.6.937.

[23]

P. E. Kloeden, P. Marín-Rubio and J. Real, Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations, Commun. Pure Appl. Anal., 8 (2009), 785-802.  doi: 10.3934/cpaa.2009.8.785.

[24]

A. Kupiainen, Statistical Theories of Turbulence, In advances in Mathematical Sciences and Applications. Gakkotosho, Tokyo, 2003.

[25]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux limites Non linéaires, Dunod, Paris, 1969.

[26]

P. Marín-Rubio, A. M. Márquez-Durán and J. Real, On the convergence of solutions of globally modified Navier-Stokes equations with delays to solutions of Navier-Stokes equations with delays, Adv. Nonlinear Stud., 11 (2011), 917-927.  doi: 10.1515/ans-2011-0409.

[27]

P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Three dimensional system of globally modified Navier-Stokes equations with infinite delays, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 655-673.  doi: 10.3934/dcdsb.2010.14.655.

[28]

P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Pullback attractors for globally modified Navier-Stokes equations with infinite delays, Discrete Contin. Dyn. Syst. Ser. A, 31 (2011), 779-796.  doi: 10.3934/dcds.2011.31.779.

[29]

P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Asymptotic behavior of solutions for a three dimensional system of globally modified Navier-Stokes equations with a locally Lipschitz delay term, Nonlinear Anal., 79 (2013), 68-79.  doi: 10.1016/j.na.2012.11.006.

[30]

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

[31]

C. Prévȏt and M. Röckner, A concise Course on Stochastic Partial Differential Equations, Springer-Verlag, 2007.

[32] J. C. Robinson, Infinite-dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001.  doi: 10.1007/978-94-010-0732-0.
[33]

M. Romito, The uniqueness of weak solution of the globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 9 (2009), 425-427.  doi: 10.1515/ans-2009-0209.

[34]

M. Röckner and T. Zhang, Stochastic 3D tamed Navier-Stokes equations: Existence, uniqueness and small time large deviation principles, J. Differential Equations, 252 (2012), 716-744.  doi: 10.1016/j.jde.2011.09.030.

[35]

R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam, 1977.

[36]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, Second Edition, SIAM, Philadelphia, 1995. doi: 10.1137/1.9781611970050.

show all references

References:
[1]

D. Barbato, M. Barsati, H. Bessaih and F. Flandoli, Some rigorous results on a stochastic Goy model, J. Stat. Phys., 125 (2006), 677-716.  doi: 10.1007/s10955-006-9203-y.

[2]

A. Bensoussan, Stochastic Navier-Stokes equations, Acta Appl. Math., 38 (1995), 267-304.  doi: 10.1007/BF00996149.

[3]

A. Bensoussan and R. Temam, Equations stochastiques de type Navier-Stokes, J. Funct. Anal., 13 (1973), 195-222. 

[4]

H. Bessaih and A. Millet, Large deviation principle and inviscid shell models, Electron. J. Probab., 14 (2009), 2551-2579.  doi: 10.1214/EJP.v14-719.

[5]

H. Bessaih, F. Flandoli and E. S. Titi, Stochastic attractors for shell phenomenological models of turbulence, J. Stat. Phys., 140 (2010), 688-717.  doi: 10.1007/s10955-010-0010-0.

[6]

T. Caraballo, P. E. Kloeden and J. Real, Unique strong solutions and V-attractors of a 3- dimensional system of globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 6 (2006), 411-436.  doi: 10.1515/ans-2006-0304.

[7]

T. Caraballo and P. E. Kloeden, The three-dimensional globally modified Navier-Stokes equations: recent developments, Recent trends in Dynamical Systems, Springer Proc. Math. Stat., 35, 473-492 Springer, Basel, 2013. doi: 10.1007/978-3-0348-0451-6_18.

[8]

I. Chueshov and A. Millet, Stochastic 2D hydrodynamical type systems: well posedness and large deviations, Appl. Math. Optim., 61 (2010), 379-420.  doi: 10.1007/s00245-009-9091-z.

[9]

P. Constantin, Near Identity Transformations for the Navier-Stokes Equations, in Handbook of Mathematical Fluid Dynamics, Vol. Ⅱ, 117-141, North-Holland, Amsterdam, 2003. doi: 10.1016/S1874-5792(03)80006-X.

[10]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its applications, vol. 44, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.

[11]

A. Debussche, N. Glatt-Holtz and R. Temam, Local martingale and pathwise solutions for an abstract fluids model, Phys. D, 240 (2011), 1123-1144.  doi: 10.1016/j.physd.2011.03.009.

[12]

G. Deugoué and J. K. Djoko, On the time discretization for the globally modified 3- dimensional Navier-Stokes equations, J. Comput. Appl. Math., 235 (2011), 2015-2029.  doi: 10.1016/j.cam.2010.10.003.

[13]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 307-391.  doi: 10.1007/BF01192467.

[14]

F. Flandoli, An introduction to 3d stochastic fluid dynamics, SPDE in Hydrodynamic: Recent Progress and Prospects, Lecture Notes in Mathematics, vol. 1992, 51-150, Springer Berlin, Heidelberg, 2008. doi: 10. 1007/978-3-540-78493-7_2.

[15]

F. Flandoli and B. Maslowski, Ergodicity of the 2D Navier-Stokes equation under random perturbations, Commun. Math. Phys., 172 (1995), 119-141. 

[16]

F. Flandoli, M. Gubinelli, M. Hairer and M. Romito, Rigorous remarks about scaling laws in turbulent fluid, Commun. Math. Phys., 278 (2008), 1-29.  doi: 10.1007/s00220-007-0398-9.

[17] I. I. Gikhman and A. V. Skorohod, Stochastic Differential Equations, Springer-Verlag, Berlin, 1972. 
[18]

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

[19]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Second edition, North-Holland, Kodansha, 1989.

[20]

A. Jakubowski, The almost sure Skorokhod representation for subsequences in nonmetric spaces, translated from Teor. Veroyatnost. i Primenen, 42 (1997), 209-216.  doi: 10.1137/S0040585X97976052.

[21]

P. E. Kloeden and J. Valero, The weak connectedness of the attainability set of weak solutions of the three-dimensional Navier-Stokes equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci, 463 (2007), 1491-1508.  doi: 10.1098/rspa.2007.1831.

[22]

P. E. Kloeden, J. A. Langa and J. Real, Pullback V-attractors of the three dimensional globally modified Navier-Stokes equations: existence and finite fractal dimension, Commun. Pure Appl. Anal., 6 (2007), 937-955.  doi: 10.3934/cpaa.2007.6.937.

[23]

P. E. Kloeden, P. Marín-Rubio and J. Real, Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations, Commun. Pure Appl. Anal., 8 (2009), 785-802.  doi: 10.3934/cpaa.2009.8.785.

[24]

A. Kupiainen, Statistical Theories of Turbulence, In advances in Mathematical Sciences and Applications. Gakkotosho, Tokyo, 2003.

[25]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux limites Non linéaires, Dunod, Paris, 1969.

[26]

P. Marín-Rubio, A. M. Márquez-Durán and J. Real, On the convergence of solutions of globally modified Navier-Stokes equations with delays to solutions of Navier-Stokes equations with delays, Adv. Nonlinear Stud., 11 (2011), 917-927.  doi: 10.1515/ans-2011-0409.

[27]

P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Three dimensional system of globally modified Navier-Stokes equations with infinite delays, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 655-673.  doi: 10.3934/dcdsb.2010.14.655.

[28]

P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Pullback attractors for globally modified Navier-Stokes equations with infinite delays, Discrete Contin. Dyn. Syst. Ser. A, 31 (2011), 779-796.  doi: 10.3934/dcds.2011.31.779.

[29]

P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Asymptotic behavior of solutions for a three dimensional system of globally modified Navier-Stokes equations with a locally Lipschitz delay term, Nonlinear Anal., 79 (2013), 68-79.  doi: 10.1016/j.na.2012.11.006.

[30]

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

[31]

C. Prévȏt and M. Röckner, A concise Course on Stochastic Partial Differential Equations, Springer-Verlag, 2007.

[32] J. C. Robinson, Infinite-dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001.  doi: 10.1007/978-94-010-0732-0.
[33]

M. Romito, The uniqueness of weak solution of the globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 9 (2009), 425-427.  doi: 10.1515/ans-2009-0209.

[34]

M. Röckner and T. Zhang, Stochastic 3D tamed Navier-Stokes equations: Existence, uniqueness and small time large deviation principles, J. Differential Equations, 252 (2012), 716-744.  doi: 10.1016/j.jde.2011.09.030.

[35]

R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam, 1977.

[36]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, Second Edition, SIAM, Philadelphia, 1995. doi: 10.1137/1.9781611970050.

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