American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains
  • CPAA Home
  • This Issue
  • Next Article
    Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency
December  2020, 19(12): 5337-5365. doi: 10.3934/cpaa.2020241

Approximations of stochastic 3D tamed Navier-Stokes equations

Xuhui Peng 1, and  Rangrang Zhang 2,,

1. 

MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Hunan, 410081, China
2. 

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China

* Corresponding author

Received  January 2020 Revised  July 2020 Published  December 2020 Early access  September 2020

Fund Project: This work was supported by NSFC (No. 11801032, 11971227, 11501195, 11871476). Hunan Provincial Natural Science Foundation of China (No. 2019JJ50377). Key Laboratory of Random Complex Structures and Data Science, Academy of Mathematics and Systems Science, CAS (No. 2008DP173182). Beijing Institute of Technology Research Fund Program for Young Scholars. The Construct Program of the Key Discipline in Hunan Province

In this paper, we are concerned with 3D tamed Navier-Stokes equations with periodic boundary conditions, which can be viewed as an approximation of the classical 3D Navier-Stokes equations. We show that the strong solution of 3D tamed Navier-Stokes equations driven by Poisson random measure converges weakly to the strong solution of 3D tamed Navier-Stokes equations driven by Gaussian noise on the state space $ \mathcal{D}([0, T];\mathbb{H}^1) $.

Keywords: 3D tamed Navier-Stokes equations, strong solution, Gaussian noise, Poisson random measure, approximations.
Mathematics Subject Classification: Primary: 60H15; Secondary: 93E20, 35R60.
Citation: Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241
References:
[1]

D. Aldous, Stopping times and tightness, Ann. Probab., 6 (1978), 335-340.  doi: 10.1214/aop/1176995579.  Google Scholar

[2]

P. Billingsley, Convergence of Probability Measure, 2$^{nd}$ edition, John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9780470316962.  Google Scholar

[3]

Z. Brzeźniak and G. Dhariwal, Stochastic Tamed Navier-Stokes Equations on $\mathbb{R}^3$: The Existence and the Uniqueness of Solutions and the Existence of an Invariant Measure, J. Math. Fluid Mech., 22 (2020), 54 pp. doi: 10.1007/s00021-020-0480-z.  Google Scholar

[4]

Z. Brzeźniak, E. Hausenblas and J. Zhu, 2D Navier-Stokes equations driven by jump noise, Nonlinear Anal., 79 (2013), 122-139.  doi: 10.1016/j.na.2012.10.011.  Google Scholar

[5]

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

[6]

A. Budhiraja, P. Dupuis and V. Maroulas, Variational representations for continuous time processes, Ann. Inst. Henri Poincaré Probab. Stat., 47 (2011), 725-747.  doi: 10.1214/10-AIHP382.  Google Scholar

[7]

Z. Brzézniak, W. Liu and J. Zhu, Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise, Nonlinear Anal. Real World Appl., 17 (2014), 283-310.  doi: 10.1016/j.nonrwa.2013.12.005.  Google Scholar

[8]

G. Di Nunno and T. Zhang, Approximations of stochastic partial differential equations, Ann. Appl. Probab., 26 (2016), 1443-1466.  doi: 10.1214/15-AAP1122.  Google Scholar

[9]

Z. Dong, J. Xiong, J. Zhai and T. Zhang, A moderate deviation principle for 2-D stochastic Navier-Stokes equations driven by multiplicative Lévy noises, J. Funct. Anal., 272 (2017), 227-254.  doi: 10.1016/j.jfa.2016.10.012.  Google Scholar

[10]

Z. Dong and R. Zhang, 3D tamed Navier-Stokes equations driven by multiplicative Lévy noise: Existence, uniqueness and large deviations, J. Math. Anal. Appl., 492 (2020), 124404. doi: 10.1016/j.jmaa.2020.124404.  Google Scholar

[11]

R. Durrett, Probability: Theory and Examples, 4$^{nd}$ edition, Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9780511779398.  Google Scholar

[12]

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

[13]

A. Ichikawa, Some inequalities for martingales and stochastic convolutions, Stoch. Anal. Appl., 4 (1986), 329-339. doi: 10.1080/07362998608809094.  Google Scholar

[14]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2$^{nd}$ edition, North-Holland Mathematical Library, 1989.  Google Scholar

[15]

O. Kallenberg, Foundations of Modern Probability, 2$^{nd}$ edition, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-4015-8.  Google Scholar

[16]

P. L. Lions, Mathematical Topics in Fluid Mechanics, Clarendon Press, Oxford, 1996.  Google Scholar

[17]

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

[18]

M. T. Mohan, K. Sakthivel and S. S. Sritharan, Ergodicity for the 3D stochastic Navier-Stokes equations perturbed by Lévy noise, Math. Nachr., 292 (2019), 1056-1088.  doi: 10.1002/mana.201700339.  Google Scholar

[19]

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

[20]

M. Röckner, T. Zhang and X. Zhang, Large deviations for stochastic tamed 3D Navier-Stokes equations, Appl. Math. Optim., 61 (2010), 267-285.  doi: 10.1007/s00245-009-9089-6.  Google Scholar

[21]

M. Röckner and X. Zhang, Tamed 3D Navier-Stokes equation: existence, uniqueness and regularity, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 12 (2009), 525-549.  doi: 10.1142/S0219025709003859.  Google Scholar

[22]

M. Röckner and X. Zhang, Stochastic tamed 3D Navier-Stokes equations: existence, uniqueness and ergodicity, Probab. Theory Related Fields, 145 (2009), 211-267.  doi: 10.1007/s00440-008-0167-5.  Google Scholar

[23]

B. L. Rozovskii, Stochastic Evolution Systems, Kluwer Academic, Dordrecht, 1990. doi: 10.1007/978-94-011-3830-7.  Google Scholar

[24]

B. Schmalfuss, Qualitative properties for the stochastic Navier-Stokes equation, Nonlinear Anal., 28 (1997), 1545-1563.  doi: 10.1016/S0362-546X(96)00015-6.  Google Scholar

[25]

S. Shang and T. Zhang, Approximations of stochastic Navier-Stokes equations, Stochastic Process. Appl., 130 (2020), 2407-2432.  doi: 10.1016/j.spa.2019.07.007.  Google Scholar

[26]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. doi: https://doi.org/10.1115/1.3424338.  Google Scholar

[27]

X. Zhang, A tamed 3D Navier-Stokes equation in uniform $C^2-$domains, Nonlinear Anal., 71 (2009), 3093-3112.  doi: 10.1016/j.na.2009.01.221.  Google Scholar

show all references

References:
[1]

D. Aldous, Stopping times and tightness, Ann. Probab., 6 (1978), 335-340.  doi: 10.1214/aop/1176995579.  Google Scholar

[2]

P. Billingsley, Convergence of Probability Measure, 2$^{nd}$ edition, John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9780470316962.  Google Scholar

[3]

Z. Brzeźniak and G. Dhariwal, Stochastic Tamed Navier-Stokes Equations on $\mathbb{R}^3$: The Existence and the Uniqueness of Solutions and the Existence of an Invariant Measure, J. Math. Fluid Mech., 22 (2020), 54 pp. doi: 10.1007/s00021-020-0480-z.  Google Scholar

[4]

Z. Brzeźniak, E. Hausenblas and J. Zhu, 2D Navier-Stokes equations driven by jump noise, Nonlinear Anal., 79 (2013), 122-139.  doi: 10.1016/j.na.2012.10.011.  Google Scholar

[5]

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

[6]

A. Budhiraja, P. Dupuis and V. Maroulas, Variational representations for continuous time processes, Ann. Inst. Henri Poincaré Probab. Stat., 47 (2011), 725-747.  doi: 10.1214/10-AIHP382.  Google Scholar

[7]

Z. Brzézniak, W. Liu and J. Zhu, Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise, Nonlinear Anal. Real World Appl., 17 (2014), 283-310.  doi: 10.1016/j.nonrwa.2013.12.005.  Google Scholar

[8]

G. Di Nunno and T. Zhang, Approximations of stochastic partial differential equations, Ann. Appl. Probab., 26 (2016), 1443-1466.  doi: 10.1214/15-AAP1122.  Google Scholar

[9]

Z. Dong, J. Xiong, J. Zhai and T. Zhang, A moderate deviation principle for 2-D stochastic Navier-Stokes equations driven by multiplicative Lévy noises, J. Funct. Anal., 272 (2017), 227-254.  doi: 10.1016/j.jfa.2016.10.012.  Google Scholar

[10]

Z. Dong and R. Zhang, 3D tamed Navier-Stokes equations driven by multiplicative Lévy noise: Existence, uniqueness and large deviations, J. Math. Anal. Appl., 492 (2020), 124404. doi: 10.1016/j.jmaa.2020.124404.  Google Scholar

[11]

R. Durrett, Probability: Theory and Examples, 4$^{nd}$ edition, Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9780511779398.  Google Scholar

[12]

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

[13]

A. Ichikawa, Some inequalities for martingales and stochastic convolutions, Stoch. Anal. Appl., 4 (1986), 329-339. doi: 10.1080/07362998608809094.  Google Scholar

[14]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2$^{nd}$ edition, North-Holland Mathematical Library, 1989.  Google Scholar

[15]

O. Kallenberg, Foundations of Modern Probability, 2$^{nd}$ edition, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-4015-8.  Google Scholar

[16]

P. L. Lions, Mathematical Topics in Fluid Mechanics, Clarendon Press, Oxford, 1996.  Google Scholar

[17]

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

[18]

M. T. Mohan, K. Sakthivel and S. S. Sritharan, Ergodicity for the 3D stochastic Navier-Stokes equations perturbed by Lévy noise, Math. Nachr., 292 (2019), 1056-1088.  doi: 10.1002/mana.201700339.  Google Scholar

[19]

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

[20]

M. Röckner, T. Zhang and X. Zhang, Large deviations for stochastic tamed 3D Navier-Stokes equations, Appl. Math. Optim., 61 (2010), 267-285.  doi: 10.1007/s00245-009-9089-6.  Google Scholar

[21]

M. Röckner and X. Zhang, Tamed 3D Navier-Stokes equation: existence, uniqueness and regularity, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 12 (2009), 525-549.  doi: 10.1142/S0219025709003859.  Google Scholar

[22]

M. Röckner and X. Zhang, Stochastic tamed 3D Navier-Stokes equations: existence, uniqueness and ergodicity, Probab. Theory Related Fields, 145 (2009), 211-267.  doi: 10.1007/s00440-008-0167-5.  Google Scholar

[23]

B. L. Rozovskii, Stochastic Evolution Systems, Kluwer Academic, Dordrecht, 1990. doi: 10.1007/978-94-011-3830-7.  Google Scholar

[24]

B. Schmalfuss, Qualitative properties for the stochastic Navier-Stokes equation, Nonlinear Anal., 28 (1997), 1545-1563.  doi: 10.1016/S0362-546X(96)00015-6.  Google Scholar

[25]

S. Shang and T. Zhang, Approximations of stochastic Navier-Stokes equations, Stochastic Process. Appl., 130 (2020), 2407-2432.  doi: 10.1016/j.spa.2019.07.007.  Google Scholar

[26]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. doi: https://doi.org/10.1115/1.3424338.  Google Scholar

[27]

X. Zhang, A tamed 3D Navier-Stokes equation in uniform $C^2-$domains, Nonlinear Anal., 71 (2009), 3093-3112.  doi: 10.1016/j.na.2009.01.221.  Google Scholar

[1]

Lihuai Du, Ting Zhang. Local and global strong solution to the stochastic 3-D incompressible anisotropic Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4745-4765. doi: 10.3934/dcds.2018209
[2]

Chongsheng Cao. Sufficient conditions for the regularity to the 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1141-1151. doi: 10.3934/dcds.2010.26.1141
[3]

Xuanji Jia, Zaihong Jiang. An anisotropic regularity criterion for the 3D Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1299-1306. doi: 10.3934/cpaa.2013.12.1299
[4]

Hui Chen, Daoyuan Fang, Ting Zhang. Regularity of 3D axisymmetric Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 1923-1939. doi: 10.3934/dcds.2017081
[5]

Donatella Donatelli, Nóra Juhász. The primitive equations of the polluted atmosphere as a weak and strong limit of the 3D Navier-Stokes equations in downwind-matching coordinates. Discrete & Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022002
[6]

Teng Wang, Yi Wang. Large-time behaviors of the solution to 3D compressible Navier-Stokes equations in half space with Navier boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (7&8) : 2811-2838. doi: 10.3934/cpaa.2021080
[7]

Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163
[8]

Anthony Suen. Existence and a blow-up criterion of solution to the 3D compressible Navier-Stokes-Poisson equations with finite energy. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1775-1798. doi: 10.3934/dcds.2020093
[9]

Alessio Falocchi, Filippo Gazzola. Regularity for the 3D evolution Navier-Stokes equations under Navier boundary conditions in some Lipschitz domains. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021151
[10]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637
[11]

Shijin Ding, Zhilin Lin, Dongjuan Niu. Boundary layer for 3D plane parallel channel flows of nonhomogeneous incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2020, 40 (8) : 4579-4596. doi: 10.3934/dcds.2020193
[12]

Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142
[13]

Daoyuan Fang, Chenyin Qian. Regularity criterion for 3D Navier-Stokes equations in Besov spaces. Communications on Pure & Applied Analysis, 2014, 13 (2) : 585-603. doi: 10.3934/cpaa.2014.13.585
[14]

Yong Yang, Bingsheng Zhang. On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations:Ⅰ. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2339-2350. doi: 10.3934/dcdsb.2017101
[15]

Hakima Bessaih, María J. Garrido-Atienza. Longtime behavior for 3D Navier-Stokes equations with constant delays. Communications on Pure & Applied Analysis, 2020, 19 (4) : 1931-1948. doi: 10.3934/cpaa.2020085
[16]

Zoran Grujić. Regularity of forward-in-time self-similar solutions to the 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2006, 14 (4) : 837-843. doi: 10.3934/dcds.2006.14.837
[17]

G. Deugoué, T. Tachim Medjo. The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2593-2621. doi: 10.3934/cpaa.2018123
[18]

Fan Wu. Conditional regularity for the 3D Navier-Stokes equations in terms of the middle eigenvalue of the strain tensor. Evolution Equations & Control Theory, 2021, 10 (3) : 511-518. doi: 10.3934/eect.2020078
[19]

Pan Zhang, Lan Huang, Rui Lu, Xin-Guang Yang. Pullback dynamics of a 3D modified Navier-Stokes equations with double delays. Electronic Research Archive, 2021, 29 (6) : 4137-4157. doi: 10.3934/era.2021076
[20]

Wei Shi, Xiaona Cui, Xuezhi Li, Xin-Guang Yang. Dynamics for the 3D incompressible Navier-Stokes equations with double time delays and damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021284

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (211)
  • HTML views (100)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy