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A note on global existence for the Zakharov system on $ \mathbb{T} $
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 |
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 $).
References:
[1] |
H. Ali,
On a critical Leray-$\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. Barbato, H. 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. Barbato, F. 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. Bessaih, E. 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. Brzeźniak, B. 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. Budhiraja, P. 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. Chepyzhov, E. 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. Cheskidov, D. D. Holm, E. 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. Dong, J. 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. Fernando, E. 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. Google Scholar |
[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. Google Scholar |
[34] |
Y. Li, Y. 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. Liu, M. 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. Röckner, X. 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. Röckner, R.-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. Barbato, H. 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. Barbato, F. 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. Bessaih, E. 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. Brzeźniak, B. 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. Budhiraja, P. 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. Chepyzhov, E. 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. Cheskidov, D. D. Holm, E. 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. Dong, J. 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. Fernando, E. 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. Google Scholar |
[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. Google Scholar |
[34] |
Y. Li, Y. 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. Liu, M. 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. Röckner, X. 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. Röckner, R.-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.
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