-
Previous Article
Ground states for Kirchhoff-type equations with critical growth
- CPAA Home
- This Issue
-
Next Article
A free boundary problem for a class of parabolic-elliptic type chemotaxis model
The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior
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 |
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. |
[1] |
Tomás Caraballo, Peter E. Kloeden, José Real. Invariant measures and Statistical solutions of the globally modified Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2008, 10 (4) : 761-781. doi: 10.3934/dcdsb.2008.10.761 |
[2] |
Jochen Merker. Strong solutions of doubly nonlinear Navier-Stokes equations. Conference Publications, 2011, 2011 (Special) : 1052-1060. doi: 10.3934/proc.2011.2011.1052 |
[3] |
P.E. Kloeden, Pedro Marín-Rubio, José Real. Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations. Communications on Pure and Applied Analysis, 2009, 8 (3) : 785-802. doi: 10.3934/cpaa.2009.8.785 |
[4] |
Hamid Bellout, Jiří Neustupa, Patrick Penel. On a $\nu$-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary condition. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1353-1373. doi: 10.3934/dcds.2010.27.1353 |
[5] |
Pedro Marín-Rubio, Antonio M. Márquez-Durán, José Real. Pullback attractors for globally modified Navier-Stokes equations with infinite delays. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 779-796. doi: 10.3934/dcds.2011.31.779 |
[6] |
Zhiting Ma. Navier-Stokes limit of globally hyperbolic moment equations. Kinetic and Related Models, 2021, 14 (1) : 175-197. doi: 10.3934/krm.2021001 |
[7] |
Lihuai Du, Ting Zhang. Local and global strong solution to the stochastic 3-D incompressible anisotropic Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4745-4765. doi: 10.3934/dcds.2018209 |
[8] |
Michele Campiti, Giovanni P. Galdi, Matthias Hieber. Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1613-1627. doi: 10.3934/cpaa.2014.13.1613 |
[9] |
Kumarasamy Sakthivel, Sivaguru S. Sritharan. Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise. Evolution Equations and Control Theory, 2012, 1 (2) : 355-392. doi: 10.3934/eect.2012.1.355 |
[10] |
Takeshi Taniguchi. The existence and decay estimates of the solutions to $3$D stochastic Navier-Stokes equations with additive noise in an exterior domain. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4323-4341. doi: 10.3934/dcds.2014.34.4323 |
[11] |
Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163 |
[12] |
Kuijie Li, Tohru Ozawa, Baoxiang Wang. Dynamical behavior for the solutions of the Navier-Stokes equation. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1511-1560. doi: 10.3934/cpaa.2018073 |
[13] |
Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure and Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747 |
[14] |
Jie Liao, Xiao-Ping Wang. Stability of an efficient Navier-Stokes solver with Navier boundary condition. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 153-171. doi: 10.3934/dcdsb.2012.17.153 |
[15] |
P.E. Kloeden, José A. Langa, José Real. Pullback V-attractors of the 3-dimensional globally modified Navier-Stokes equations. Communications on Pure and Applied Analysis, 2007, 6 (4) : 937-955. doi: 10.3934/cpaa.2007.6.937 |
[16] |
Pedro Marín-Rubio, Antonio M. Márquez-Durán, José Real. Three dimensional system of globally modified Navier-Stokes equations with infinite delays. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 655-673. doi: 10.3934/dcdsb.2010.14.655 |
[17] |
Chuong V. Tran, Theodore G. Shepherd, Han-Ru Cho. Stability of stationary solutions of the forced Navier-Stokes equations on the two-torus. Discrete and Continuous Dynamical Systems - B, 2002, 2 (4) : 483-494. doi: 10.3934/dcdsb.2002.2.483 |
[18] |
Ana Bela Cruzeiro. Navier-Stokes and stochastic Navier-Stokes equations via Lagrange multipliers. Journal of Geometric Mechanics, 2019, 11 (4) : 553-560. doi: 10.3934/jgm.2019027 |
[19] |
Matthew Paddick. The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2673-2709. doi: 10.3934/dcds.2016.36.2673 |
[20] |
Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]