Large deviations for stochastic 3D Leray-<inline-formula><tex-math id="M1">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> model with fractional dissipation

Shihu Li; Wei Liu; Yingchao Xie

doi:10.3934/cpaa.2019113

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

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

Shihu Li ^1, , Wei Liu ^2,, and Yingchao Xie ^2,

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 $).

Keywords: Large deviation principle, Leray-$ \alpha $ model, fractional Laplacian, Navier-Stokes equation, weak convergence approach.

Mathematics Subject Classification: Primary: 60H15, 60F10; Secondary: 35Q30, 35R11.

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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[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. Google Scholar
[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. Google Scholar
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[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. Google Scholar
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[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[26]	B. Ferrario, Characterization of the law for 3D stochastic hyperviscous fluids, Electron. J. Probab., 21 (2016), 1-22. doi: 10.1214/16-EJP4607. Google Scholar
[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. Google Scholar
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[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. Google Scholar
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[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[39]	W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Universitext, Springer, 2015. doi: 10.1007/978-3-319-22354-4. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[45]	A. A. Pukhalskii, On the theory of large deviations, Theory Probab. Appl., 38 (1993), 490-497. doi: 10.1137/1138045. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[49]	R. B. Sowers, Large deviations for a reaction-diffusion equation with non-Gaussian perturbations, Ann. Probab., 20 (1992), 504-537. Google Scholar
[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. Google Scholar
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[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. Google Scholar
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[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	A. Budhiraja and P. Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion, Probab. Math. Statist., 20 (2000), 39-61. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[21]	A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Jones and Bartlett, Boston, 1993. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[26]	B. Ferrario, Characterization of the law for 3D stochastic hyperviscous fluids, Electron. J. Probab., 21 (2016), 1-22. doi: 10.1214/16-EJP4607. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[31]	J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, (French) Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354. Google Scholar
[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. Google Scholar
[35]	J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[39]	W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Universitext, Springer, 2015. doi: 10.1007/978-3-319-22354-4. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[45]	A. A. Pukhalskii, On the theory of large deviations, Theory Probab. Appl., 38 (1993), 490-497. doi: 10.1137/1138045. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[49]	R. B. Sowers, Large deviations for a reaction-diffusion equation with non-Gaussian perturbations, Ann. Probab., 20 (1992), 504-537. Google Scholar
[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. Google Scholar
[51]	E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton, NJ: Princeton University Press, 1970. Google Scholar
[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. Google Scholar
[53]	S. R. S. Varadhan, Asymptotic probabilities and differential equations, Commun. Pure Appl. Math., 19 (1966), 261-286. doi: 10.1002/cpa.3160190303. Google Scholar
[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. Google Scholar
[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. Google Scholar
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American Institute of Mathematical Sciences

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

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

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