September 2019, 18(5): 2491-2509. doi: 10.3934/cpaa.2019113

Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation

1. 

School of Mathematical Sciences, Nankai University, 300071 Tianjin, China

2. 

School of Mathematics and Statistics, Jiangsu Normal University, 221116 Xuzhou, China

* Corresponding author

Received  June 2018 Revised  December 2018 Published  April 2019

Fund Project: The research of W. Liu is supported by NSFC (No. 11571147, 11822106, 11831014), NSF of Jiangsu Province (No. BK20160004) and the Qing Lan Project; the research of Y. Xie is supported by NSFC (No. 11771187) and PAPD of Jiangsu Higher Education Institutions

In this paper we establish the Freidlin-Wentzell's large deviation principle for stochastic 3D Leray-$ \alpha $ model with general fractional dissipation and small multiplicative noise. This model is the stochastic 3D Navier-Stokes equations regularized through a smoothing kernel of order $ \theta_1 $ in the nonlinear term and a $ \theta_2 $-fractional Laplacian. The main result generalizes the corresponding LDP result of the classical stochastic 3D Leray-$ \alpha $ model ($ \theta_1 = 1 $, $ \theta_2 = 1 $), and it is also applicable to the stochastic 3D hyperviscous Navier-Stokes equations ($ \theta_1 = 0 $, $ \theta_2\geq\frac{5}{4} $) and stochastic 3D critical Leray-$ \alpha $ model ($ \theta_1 = \frac{1}{4} $, $ \theta_2 = 1 $).

Citation: Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113
References:
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H. Ali, On a critical Leray-$\alpha$ model of turbulence, Nonlinear Anal. Real World Appl., 14 (2013), 1563-1584. doi: 10.1016/j.nonrwa.2012.10.019.

[2]

C. T. Anh and N. T. Da, The exponential behaviour and stabilizability of stochastic 2D hydrodynamical type systems, Stochastics, 89 (2017), 593-618. doi: 10.1080/17442508.2016.1269767.

[3]

D. BarbatoH. Bessaih and B. Ferrario, On a stochastic Leray-$\alpha$ model of Euler equations, Stochastic Process. Appl., 124 (2014), 199-219. doi: 10.1016/j.spa.2013.07.002.

[4]

D. BarbatoF. Morandin and M. Romito, Global regularity for a slightly supercritical hyperdissipative Navier-Stokes system, Anal. PDE, 7 (2014), 2009-2027. doi: 10.2140/apde.2014.7.2009.

[5]

V. Barbu and M. Röckner, An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise, J. Eur. Math. Soc., 17 (2015), 1789-1815. doi: 10.4171/JEMS/545.

[6]

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

[7]

H. Bessaih and B. Ferrario, The regularized 3D Boussinesq equations with fractional Laplacian and no diffusion, J. Differential Equations, 262 (2017), 1822-1849. doi: 10.1016/j.jde.2016.10.032.

[8]

H. BessaihE. Hausenblas and P. A. Razafimandimby, Strong solutions to stochastic hydrodynamical systems with multiplicative noise of jump type, Nonlinear Differ. Equ. Appl., 22 (2015), 1661-1697. doi: 10.1007/s00030-015-0339-9.

[9]

H. Bessaih and A. Millet, Large deviations and the zero viscosity limit for 2D stochastic Navier-Stokes equations with free boundary, SIAM J. Math. Anal., 44 (2012), 1861-1893. doi: 10.1137/110827235.

[10]

H. Bessaih and P. A. Razafimandimby, On the rate of convergence of the 2-D stochastic Leray-$\alpha$ model to the 2-D stochastic Navier-Stokes equations with multiplicative noise, Appl. Math. Optim., 74 (2016), 1-25. doi: 10.1007/s00245-015-9303-7.

[11]

Z. BrzeźniakB. Goldys and T. Jegaraj, Large deviations and transitions between equilibria for stochastic Landau-Lifshitz-Gilbert equation, Arch. Ration. Mech. Anal., 226 (2017), 497-558. doi: 10.1007/s00205-017-1117-0.

[12]

A. Budhiraja and P. Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion, Probab. Math. Statist., 20 (2000), 39-61.

[13]

A. BudhirajaP. Dupuis and V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems, Ann. Probab., 36 (2008), 1390-1420. doi: 10.1214/07-AOP362.

[14]

S. Cerrai and M. Röckner, Large deviations for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term, Ann. Probab., 32 (2004), 1100-1139. doi: 10.1214/aop/1079021473.

[15]

Y. Chen and H. Gao, Well-posedness and large deviations for a class of SPDEs with Lévy noise, J. Differential Equations, 263 (2017), 5216-5252. doi: 10.1016/j.jde.2017.06.016.

[16]

V. V. ChepyzhovE. S. Titi and M. I. Vishik, On the convergence of solutions of the Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system, Discrete Contin. Dyn. Syst., 17 (2007), 481-500. doi: 10.3934/dcds.2007.17.481.

[17]

A. CheskidovD. D. HolmE. Olson and E. S. Titi, On a Leray-$\alpha$ model of turbulence, Proceedings of the Royal Society A, 461 (2005), 629-649. doi: 10.1098/rspa.2004.1373.

[18]

I. Chueshov and S. Kuksin, Stochastic 3D Navier-Stokes equations in a thin domain and its $\alpha$-approximation, Phys. D, 237 (2008), 1352-1367. doi: 10.1016/j.physd.2008.03.012.

[19]

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.

[20]

L. Debbi, Well-posedness of the multidimensional fractional stochastic Navier-Stokes equations on the torus and on bounded domains, J. Math. Fluid Mech., 18 (2016), 25-69. doi: 10.1007/s00021-015-0234-5.

[21]

A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Jones and Bartlett, Boston, 1993.

[22]

G. Deugoué and M. Sango, On the strong solution for the 3D stochastic Leray-alpha model, Bound. Value Probl., (2010), Art. ID 723018, 31pp. doi: 10.1155/2010/723018.

[23]

Z. DongJ. Zhai and R. Zhang, Large deviation principles for 3D stochastic primitive equations, J. Differential Equations, 263 (2017), 3110-3146. doi: 10.1016/j.jde.2017.04.025.

[24]

J. Duan and A. Millet, Large deviations for the Boussinesq equations under random influences, Stochastic Process. Appl., 119 (2009), 2052-2081. doi: 10.1016/j.spa.2008.10.004.

[25]

P. W. FernandoE. Hausenblas and P. A. Razafimandimby, Irreducibility and exponential mixing of some stochastic hydrodynamical systems driven by pure jump noise, Comm. Math. Phys., 348 (2016), 535-565. doi: 10.1007/s00220-016-2693-9.

[26]

B. Ferrario, Characterization of the law for 3D stochastic hyperviscous fluids, Electron. J. Probab., 21 (2016), 1-22. doi: 10.1214/16-EJP4607.

[27]

F. Flandoli, A stochastic view over the open problem of well-posedness for the 3D Navier-Stokes equations. Stochastic analysis: a series of lectures, Progr. Probab., 68 (2015), 221-246. doi: 10.1007/978-3-0348-0909-2_8.

[28]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, 3$^{nd}$edition, Grundlehren der mathematischen Wissenschaften, 260. Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-25847-3.

[29]

N. Glatt-Holtz and V. Vicol, Local and global existence of smooth solutions for the stochastic Euler equations with multiplicative noise, Ann. Probab., 42 (2014), 80-145. doi: 10.1214/12-AOP773.

[30]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.

[31]

J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, (French) Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.

[32]

S. Li, W. Liu and Y. Xie, Stochastic 3D Leray-$\alpha$ model with fractional dissipation.

[33]

S. Li, W. Liu and Y. Xie, Ergodicity of 3D Leray-$\alpha$ model with fractional dissipation and degenerate stochastic forcing, Infin. Dimens. Anal. Quantum Probab. Relat. Top., In press.

[34]

Y. LiY. Xie and X. Zhang, Large deviation principle for stochastic heat equation with memory, Discrete Contin. Dyn. Syst.-A, 35 (2015), 5221-5237. doi: 10.3934/dcds.2015.35.5221.

[35]

J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969.

[36]

W. Liu, Large deviations for stochastic evolution equations with small multiplicative noise, Appl. Math. Optim., 61 (2010), 27-56. doi: 10.1007/s00245-009-9072-2.

[37]

W. Liu, Existence and uniqueness of solutions to nonlinear evolution equations with locally monotone operators, Nonlinear Anal., 74 (2011), 7543-7561. doi: 10.1016/j.na.2011.08.018.

[38]

W. Liu and M. Röckner, Local and global well-posedness of SPDE with generalized coercivity conditions, J. Differential Equations, 254 (2013), 725-755. doi: 10.1016/j.jde.2012.09.014.

[39]

W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Universitext, Springer, 2015. doi: 10.1007/978-3-319-22354-4.

[40]

W. LiuM. Röckner and X.-C. Zhu, Large deviation principles for the stochastic quasi-geostrophic equations, Stochastic Process. Appl., 123 (2013), 3299-3327. doi: 10.1016/j.spa.2013.03.020.

[41]

W. Liu, C. Tao and J. Zhu, Large deviation principle for a class of SPDE with locally monotone coefficients, Sci. China Math., In press. doi: 10.1007/s11425-018-9440-3.

[42]

D. Martirosyan, Large deviations for stationary measures of stochastic nonlinear wave equations with smooth white noise, Comm. Pure Appl. Math., 70 (2017), 1754-1797. doi: 10.1002/cpa.21693.

[43]

E. Olson and E. S. Titi, Viscosity versus vorticity stretching: global well-posedness for a family of Navier-Stokes-alpha-like models, Nonlinear Anal., 6 (2007), 2427-2458. doi: 10.1016/j.na.2006.03.030.

[44]

N. Pennington, Global solutions to the generalized Leray-alpha equation with mixed dissipation terms, Nonlinear Anal., 136 (2016), 102-116. doi: 10.1016/j.na.2016.02.006.

[45]

A. A. Pukhalskii, On the theory of large deviations, Theory Probab. Appl., 38 (1993), 490-497. doi: 10.1137/1138045.

[46]

J. Ren and X. Zhang, Freidlin-Wentzell's large deviations for stochastic evolution equations, J. Funct. Anal., 254 (2008), 3148-3172. doi: 10.1016/j.jfa.2008.02.010.

[47]

M. RöcknerX. Zhang and T. S. Zhang, Large deviations for stochastic tamed 3D Navier-Stokes equations, Appl. Math. Optim., 61 (2010), 267-85. doi: 10.1007/s00245-009-9089-6.

[48]

M. RöcknerR.-C. Zhu and X.-C. Zhu, Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise, Stochastic Process. Appl., 124 (2014), 1974-2002. doi: 10.1016/j.spa.2014.01.010.

[49]

R. B. Sowers, Large deviations for a reaction-diffusion equation with non-Gaussian perturbations, Ann. Probab., 20 (1992), 504-537.

[50]

S. S. Sritharan and P. Sundar, Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise, Stochess Process. Appl., 116 (2006), 1636-1659. doi: 10.1016/j.spa.2006.04.001.

[51] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton, NJ: Princeton University Press, 1970.
[52]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, vol.2, North-Holland Publishing Co., Amsterdam-New York, 1979.

[53]

S. R. S. Varadhan, Asymptotic probabilities and differential equations, Commun. Pure Appl. Math., 19 (1966), 261-286. doi: 10.1002/cpa.3160190303.

[54]

F.-Y. Wang and L. Xu, Derivative formula and applications for hyperdissipative stochastic Navier-Stokes/Burgers equations, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 15 (2012), 1250020, 19pp. doi: 10.1142/S0219025712500208.

[55]

J. Xiong and J. Zhai, Large deviations for locally monotone stochastic partial differential equations driven by Lévy noise, Bernoulli, 24 (2018), 2842-2874. doi: 10.3150/17-BEJ947.

[56]

K. Yamazaki, On the global regularity of generalized Leray-alpha type models, Nonlinear Anal., 75 (2012), 503-515. doi: 10.1016/j.na.2011.08.051.

[57]

J. Yang and J. Zhai, Asymptotics of stochastic 2D hydrodynamical type systems in unbounded domains, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 20 (2017), 1750017, 25pp. doi: 10.1142/S0219025717500175.

[58]

J. Zhai and T. S. Zhang, Large deviations for stochastic models of two-dimensional second grade fluids, Appl. Math. Optim., 75 (2017), 471-498. doi: 10.1007/s00245-016-9338-4.

show all references

References:
[1]

H. Ali, On a critical Leray-$\alpha$ model of turbulence, Nonlinear Anal. Real World Appl., 14 (2013), 1563-1584. doi: 10.1016/j.nonrwa.2012.10.019.

[2]

C. T. Anh and N. T. Da, The exponential behaviour and stabilizability of stochastic 2D hydrodynamical type systems, Stochastics, 89 (2017), 593-618. doi: 10.1080/17442508.2016.1269767.

[3]

D. BarbatoH. Bessaih and B. Ferrario, On a stochastic Leray-$\alpha$ model of Euler equations, Stochastic Process. Appl., 124 (2014), 199-219. doi: 10.1016/j.spa.2013.07.002.

[4]

D. BarbatoF. Morandin and M. Romito, Global regularity for a slightly supercritical hyperdissipative Navier-Stokes system, Anal. PDE, 7 (2014), 2009-2027. doi: 10.2140/apde.2014.7.2009.

[5]

V. Barbu and M. Röckner, An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise, J. Eur. Math. Soc., 17 (2015), 1789-1815. doi: 10.4171/JEMS/545.

[6]

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

[7]

H. Bessaih and B. Ferrario, The regularized 3D Boussinesq equations with fractional Laplacian and no diffusion, J. Differential Equations, 262 (2017), 1822-1849. doi: 10.1016/j.jde.2016.10.032.

[8]

H. BessaihE. Hausenblas and P. A. Razafimandimby, Strong solutions to stochastic hydrodynamical systems with multiplicative noise of jump type, Nonlinear Differ. Equ. Appl., 22 (2015), 1661-1697. doi: 10.1007/s00030-015-0339-9.

[9]

H. Bessaih and A. Millet, Large deviations and the zero viscosity limit for 2D stochastic Navier-Stokes equations with free boundary, SIAM J. Math. Anal., 44 (2012), 1861-1893. doi: 10.1137/110827235.

[10]

H. Bessaih and P. A. Razafimandimby, On the rate of convergence of the 2-D stochastic Leray-$\alpha$ model to the 2-D stochastic Navier-Stokes equations with multiplicative noise, Appl. Math. Optim., 74 (2016), 1-25. doi: 10.1007/s00245-015-9303-7.

[11]

Z. BrzeźniakB. Goldys and T. Jegaraj, Large deviations and transitions between equilibria for stochastic Landau-Lifshitz-Gilbert equation, Arch. Ration. Mech. Anal., 226 (2017), 497-558. doi: 10.1007/s00205-017-1117-0.

[12]

A. Budhiraja and P. Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion, Probab. Math. Statist., 20 (2000), 39-61.

[13]

A. BudhirajaP. Dupuis and V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems, Ann. Probab., 36 (2008), 1390-1420. doi: 10.1214/07-AOP362.

[14]

S. Cerrai and M. Röckner, Large deviations for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term, Ann. Probab., 32 (2004), 1100-1139. doi: 10.1214/aop/1079021473.

[15]

Y. Chen and H. Gao, Well-posedness and large deviations for a class of SPDEs with Lévy noise, J. Differential Equations, 263 (2017), 5216-5252. doi: 10.1016/j.jde.2017.06.016.

[16]

V. V. ChepyzhovE. S. Titi and M. I. Vishik, On the convergence of solutions of the Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system, Discrete Contin. Dyn. Syst., 17 (2007), 481-500. doi: 10.3934/dcds.2007.17.481.

[17]

A. CheskidovD. D. HolmE. Olson and E. S. Titi, On a Leray-$\alpha$ model of turbulence, Proceedings of the Royal Society A, 461 (2005), 629-649. doi: 10.1098/rspa.2004.1373.

[18]

I. Chueshov and S. Kuksin, Stochastic 3D Navier-Stokes equations in a thin domain and its $\alpha$-approximation, Phys. D, 237 (2008), 1352-1367. doi: 10.1016/j.physd.2008.03.012.

[19]

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.

[20]

L. Debbi, Well-posedness of the multidimensional fractional stochastic Navier-Stokes equations on the torus and on bounded domains, J. Math. Fluid Mech., 18 (2016), 25-69. doi: 10.1007/s00021-015-0234-5.

[21]

A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Jones and Bartlett, Boston, 1993.

[22]

G. Deugoué and M. Sango, On the strong solution for the 3D stochastic Leray-alpha model, Bound. Value Probl., (2010), Art. ID 723018, 31pp. doi: 10.1155/2010/723018.

[23]

Z. DongJ. Zhai and R. Zhang, Large deviation principles for 3D stochastic primitive equations, J. Differential Equations, 263 (2017), 3110-3146. doi: 10.1016/j.jde.2017.04.025.

[24]

J. Duan and A. Millet, Large deviations for the Boussinesq equations under random influences, Stochastic Process. Appl., 119 (2009), 2052-2081. doi: 10.1016/j.spa.2008.10.004.

[25]

P. W. FernandoE. Hausenblas and P. A. Razafimandimby, Irreducibility and exponential mixing of some stochastic hydrodynamical systems driven by pure jump noise, Comm. Math. Phys., 348 (2016), 535-565. doi: 10.1007/s00220-016-2693-9.

[26]

B. Ferrario, Characterization of the law for 3D stochastic hyperviscous fluids, Electron. J. Probab., 21 (2016), 1-22. doi: 10.1214/16-EJP4607.

[27]

F. Flandoli, A stochastic view over the open problem of well-posedness for the 3D Navier-Stokes equations. Stochastic analysis: a series of lectures, Progr. Probab., 68 (2015), 221-246. doi: 10.1007/978-3-0348-0909-2_8.

[28]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, 3$^{nd}$edition, Grundlehren der mathematischen Wissenschaften, 260. Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-25847-3.

[29]

N. Glatt-Holtz and V. Vicol, Local and global existence of smooth solutions for the stochastic Euler equations with multiplicative noise, Ann. Probab., 42 (2014), 80-145. doi: 10.1214/12-AOP773.

[30]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.

[31]

J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, (French) Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.

[32]

S. Li, W. Liu and Y. Xie, Stochastic 3D Leray-$\alpha$ model with fractional dissipation.

[33]

S. Li, W. Liu and Y. Xie, Ergodicity of 3D Leray-$\alpha$ model with fractional dissipation and degenerate stochastic forcing, Infin. Dimens. Anal. Quantum Probab. Relat. Top., In press.

[34]

Y. LiY. Xie and X. Zhang, Large deviation principle for stochastic heat equation with memory, Discrete Contin. Dyn. Syst.-A, 35 (2015), 5221-5237. doi: 10.3934/dcds.2015.35.5221.

[35]

J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969.

[36]

W. Liu, Large deviations for stochastic evolution equations with small multiplicative noise, Appl. Math. Optim., 61 (2010), 27-56. doi: 10.1007/s00245-009-9072-2.

[37]

W. Liu, Existence and uniqueness of solutions to nonlinear evolution equations with locally monotone operators, Nonlinear Anal., 74 (2011), 7543-7561. doi: 10.1016/j.na.2011.08.018.

[38]

W. Liu and M. Röckner, Local and global well-posedness of SPDE with generalized coercivity conditions, J. Differential Equations, 254 (2013), 725-755. doi: 10.1016/j.jde.2012.09.014.

[39]

W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Universitext, Springer, 2015. doi: 10.1007/978-3-319-22354-4.

[40]

W. LiuM. Röckner and X.-C. Zhu, Large deviation principles for the stochastic quasi-geostrophic equations, Stochastic Process. Appl., 123 (2013), 3299-3327. doi: 10.1016/j.spa.2013.03.020.

[41]

W. Liu, C. Tao and J. Zhu, Large deviation principle for a class of SPDE with locally monotone coefficients, Sci. China Math., In press. doi: 10.1007/s11425-018-9440-3.

[42]

D. Martirosyan, Large deviations for stationary measures of stochastic nonlinear wave equations with smooth white noise, Comm. Pure Appl. Math., 70 (2017), 1754-1797. doi: 10.1002/cpa.21693.

[43]

E. Olson and E. S. Titi, Viscosity versus vorticity stretching: global well-posedness for a family of Navier-Stokes-alpha-like models, Nonlinear Anal., 6 (2007), 2427-2458. doi: 10.1016/j.na.2006.03.030.

[44]

N. Pennington, Global solutions to the generalized Leray-alpha equation with mixed dissipation terms, Nonlinear Anal., 136 (2016), 102-116. doi: 10.1016/j.na.2016.02.006.

[45]

A. A. Pukhalskii, On the theory of large deviations, Theory Probab. Appl., 38 (1993), 490-497. doi: 10.1137/1138045.

[46]

J. Ren and X. Zhang, Freidlin-Wentzell's large deviations for stochastic evolution equations, J. Funct. Anal., 254 (2008), 3148-3172. doi: 10.1016/j.jfa.2008.02.010.

[47]

M. RöcknerX. Zhang and T. S. Zhang, Large deviations for stochastic tamed 3D Navier-Stokes equations, Appl. Math. Optim., 61 (2010), 267-85. doi: 10.1007/s00245-009-9089-6.

[48]

M. RöcknerR.-C. Zhu and X.-C. Zhu, Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise, Stochastic Process. Appl., 124 (2014), 1974-2002. doi: 10.1016/j.spa.2014.01.010.

[49]

R. B. Sowers, Large deviations for a reaction-diffusion equation with non-Gaussian perturbations, Ann. Probab., 20 (1992), 504-537.

[50]

S. S. Sritharan and P. Sundar, Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise, Stochess Process. Appl., 116 (2006), 1636-1659. doi: 10.1016/j.spa.2006.04.001.

[51] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton, NJ: Princeton University Press, 1970.
[52]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, vol.2, North-Holland Publishing Co., Amsterdam-New York, 1979.

[53]

S. R. S. Varadhan, Asymptotic probabilities and differential equations, Commun. Pure Appl. Math., 19 (1966), 261-286. doi: 10.1002/cpa.3160190303.

[54]

F.-Y. Wang and L. Xu, Derivative formula and applications for hyperdissipative stochastic Navier-Stokes/Burgers equations, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 15 (2012), 1250020, 19pp. doi: 10.1142/S0219025712500208.

[55]

J. Xiong and J. Zhai, Large deviations for locally monotone stochastic partial differential equations driven by Lévy noise, Bernoulli, 24 (2018), 2842-2874. doi: 10.3150/17-BEJ947.

[56]

K. Yamazaki, On the global regularity of generalized Leray-alpha type models, Nonlinear Anal., 75 (2012), 503-515. doi: 10.1016/j.na.2011.08.051.

[57]

J. Yang and J. Zhai, Asymptotics of stochastic 2D hydrodynamical type systems in unbounded domains, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 20 (2017), 1750017, 25pp. doi: 10.1142/S0219025717500175.

[58]

J. Zhai and T. S. Zhang, Large deviations for stochastic models of two-dimensional second grade fluids, Appl. Math. Optim., 75 (2017), 471-498. doi: 10.1007/s00245-016-9338-4.

[1]

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