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Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency
Approximations of stochastic 3D tamed Navier-Stokes equations
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 |
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) $.
References:
[1] |
D. Aldous,
Stopping times and tightness, Ann. Probab., 6 (1978), 335-340.
doi: 10.1214/aop/1176995579. |
[2] |
P. Billingsley, Convergence of Probability Measure, 2$^{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. |
show all references
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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