Approximations of stochastic 3D tamed Navier-Stokes equations

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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 and 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.
[2]	P. Billingsley, Convergence of Probability Measure, 2$^{nd}$ edition, John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9780470316962.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[11]	R. Durrett, Probability: Theory and Examples, 4$^{nd}$ edition, Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9780511779398.
[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.
[13]	A. Ichikawa, Some inequalities for martingales and stochastic convolutions, Stoch. Anal. Appl., 4 (1986), 329-339. doi: 10.1080/07362998608809094.
[14]	N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2$^{nd}$ edition, North-Holland Mathematical Library, 1989.
[15]	O. Kallenberg, Foundations of Modern Probability, 2$^{nd}$ edition, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-4015-8.
[16]	P. L. Lions, Mathematical Topics in Fluid Mechanics, Clarendon Press, Oxford, 1996.
[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.
[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.
[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.
[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.
[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.
[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.
[23]	B. L. Rozovskii, Stochastic Evolution Systems, Kluwer Academic, Dordrecht, 1990. doi: 10.1007/978-94-011-3830-7.
[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.
[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.
[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.
[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.

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References:

[1]	D. Aldous, Stopping times and tightness, Ann. Probab., 6 (1978), 335-340. doi: 10.1214/aop/1176995579.
[2]	P. Billingsley, Convergence of Probability Measure, 2$^{nd}$ edition, John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9780470316962.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[11]	R. Durrett, Probability: Theory and Examples, 4$^{nd}$ edition, Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9780511779398.
[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.
[13]	A. Ichikawa, Some inequalities for martingales and stochastic convolutions, Stoch. Anal. Appl., 4 (1986), 329-339. doi: 10.1080/07362998608809094.
[14]	N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2$^{nd}$ edition, North-Holland Mathematical Library, 1989.
[15]	O. Kallenberg, Foundations of Modern Probability, 2$^{nd}$ edition, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-4015-8.
[16]	P. L. Lions, Mathematical Topics in Fluid Mechanics, Clarendon Press, Oxford, 1996.
[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.
[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.
[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.
[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.
[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.
[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.
[23]	B. L. Rozovskii, Stochastic Evolution Systems, Kluwer Academic, Dordrecht, 1990. doi: 10.1007/978-94-011-3830-7.
[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.
[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.
[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.
[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.

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