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

G. Deugoué; T. Tachim Medjo

doi:10.3934/cpaa.2018123

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2018, Volume 17, Issue 6: 2593-2621. Doi: 10.3934/cpaa.2018123

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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
* Corresponding author

Received: September 30, 2016

Revised: June 30, 2017

Published: June 2018

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Mathematics Subject Classification: 35R60, 35Q35, 60H15, 76M35, 86A05.

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[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.