October  2022, 21(10): 3479-3498. doi: 10.3934/cpaa.2022111

Large deviations for stochastic $ 2D $ Navier-Stokes equations on time-dependent domains

School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China

*Corresponding author

Received  November 2021 Revised  February 2022 Published  October 2022 Early access  July 2022

Fund Project: This work is partially supported by NSFC (No. 12131019, 11971456, 11721101), Jianliang Zhai's research is supported by School Start-up Fund (USTC) KY0010000036, the Fundamental Research Funds for the Central Universities (No. WK3470000016)

A Freidlin-Wentzell-type large deviation principle is established for $ 2D $ stochastic Navier-Stokes equations on time-dependent domains driven by Brownian motion, which captures situations where the regions of the fluid change with time.

Citation: Wei Wang, Jianliang Zhai, Tusheng Zhang. Large deviations for stochastic $ 2D $ Navier-Stokes equations on time-dependent domains. Communications on Pure and Applied Analysis, 2022, 21 (10) : 3479-3498. doi: 10.3934/cpaa.2022111
References:
[1]

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.

[2]

D. N. Bock, On the Navier-Stokes equations in noncylindrical domains, J. Differ. Equ., 25 (1977), 151-162.  doi: 10.1016/0022-0396(77)90197-8.

[3]

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

[4]

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.

[5]

M.-H. Chang, Large deviation for Navier-Stokes equations with small stochastic perturbation, Appl. Math. Comput., 76 (1996), 65-93.  doi: 10.1016/0096-3003(95)00150-6.

[6]

P. Dupuis and R. S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations, Wiley-Interscience, New York, 1997. doi: 10.1002/9781118165904.

[7]

F. Flandoli and B. Maslowski, Ergodicity of the 2D Navier-Stokes equation under random perturbations, Commun. Math. Phys., 171 (1995), 119-141. 

[8]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Th. Rel. Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.

[9]

M. Hairer and J. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann. Math., 164 (2006), 993-1032.  doi: 10.4007/annals.2006.164.993.

[10]

A. Inoue and M. Wakimoto, On existence of solutions of the Navier-Stokes equation in a time dependent domain, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24 (1977), 303-319.

[11]

P. Kotelenez, Existence, uniqueness and smoothness for a class of function valued stochastic partial differential equations, Stochastics Stochastics Rep., 41 (1992), 177-199.  doi: 10.1080/17442509208833801.

[12]

W. Liu and M. Röckner, SPDEs in Hilbert space with locally monotone efficients, J. Funct. Anal., 259 (2010), 2902-2922.  doi: 10.1016/j.jfa.2010.05.012.

[13]

T. Miyakawa and Y. Teramoto, Existence and periodicity of weak solutions of the Navier-Stokes equations in a time dependent domain, Hiroshima Math. J., 12 (1982), 513-528. 

[14]

J. L. Menaldi and S. S. Sritharan, Stochastic 2-D Navier-Stokes equation, Appl. Math. Optim, 46 (2002), 31-53.  doi: 10.1007/s00245-002-0734-6.

[15]

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

[16]

D. Stroock, An Introduction to the Theory of Large Deviations, Springer-Verlag, Universitext, New York, 1984. doi: 10.1007/978-1-4613-8514-1.

[17]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam, 1979.

[18]

S. R. S. Varadhan, Large Deviations and its Applications, in: CBMS-NSF Series in Applied Mathematics, vol. 46, SIAM, Philadelphia, 1984. doi: 10.1137/1.9781611970241.bm.

[19]

J. B. Walsh, An introduction to stochastic partial differential equations, in École d'été de probabilités de Saint-Flour, Springer, Berlin, 1986. doi: 10.1007/BFb0074920.

[20]

R. WangJ. Zhai and T. Zhang, A moderate deviation principle for 2D stochastic Navier-Stokes equations, J. Differ. Equ., 258 (2006), 3363-3390.  doi: 10.1016/j.jde.2015.01.008.

[21]

W. Wang, J. Zhai and T. Zhang, Stochastic 2D Navier-Stokes equations on time-dependent domains, J. Theor. Probab., (2022). doi: 10.1007/s10959-021-01150-0.

show all references

References:
[1]

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.

[2]

D. N. Bock, On the Navier-Stokes equations in noncylindrical domains, J. Differ. Equ., 25 (1977), 151-162.  doi: 10.1016/0022-0396(77)90197-8.

[3]

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

[4]

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.

[5]

M.-H. Chang, Large deviation for Navier-Stokes equations with small stochastic perturbation, Appl. Math. Comput., 76 (1996), 65-93.  doi: 10.1016/0096-3003(95)00150-6.

[6]

P. Dupuis and R. S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations, Wiley-Interscience, New York, 1997. doi: 10.1002/9781118165904.

[7]

F. Flandoli and B. Maslowski, Ergodicity of the 2D Navier-Stokes equation under random perturbations, Commun. Math. Phys., 171 (1995), 119-141. 

[8]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Th. Rel. Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.

[9]

M. Hairer and J. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann. Math., 164 (2006), 993-1032.  doi: 10.4007/annals.2006.164.993.

[10]

A. Inoue and M. Wakimoto, On existence of solutions of the Navier-Stokes equation in a time dependent domain, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24 (1977), 303-319.

[11]

P. Kotelenez, Existence, uniqueness and smoothness for a class of function valued stochastic partial differential equations, Stochastics Stochastics Rep., 41 (1992), 177-199.  doi: 10.1080/17442509208833801.

[12]

W. Liu and M. Röckner, SPDEs in Hilbert space with locally monotone efficients, J. Funct. Anal., 259 (2010), 2902-2922.  doi: 10.1016/j.jfa.2010.05.012.

[13]

T. Miyakawa and Y. Teramoto, Existence and periodicity of weak solutions of the Navier-Stokes equations in a time dependent domain, Hiroshima Math. J., 12 (1982), 513-528. 

[14]

J. L. Menaldi and S. S. Sritharan, Stochastic 2-D Navier-Stokes equation, Appl. Math. Optim, 46 (2002), 31-53.  doi: 10.1007/s00245-002-0734-6.

[15]

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

[16]

D. Stroock, An Introduction to the Theory of Large Deviations, Springer-Verlag, Universitext, New York, 1984. doi: 10.1007/978-1-4613-8514-1.

[17]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam, 1979.

[18]

S. R. S. Varadhan, Large Deviations and its Applications, in: CBMS-NSF Series in Applied Mathematics, vol. 46, SIAM, Philadelphia, 1984. doi: 10.1137/1.9781611970241.bm.

[19]

J. B. Walsh, An introduction to stochastic partial differential equations, in École d'été de probabilités de Saint-Flour, Springer, Berlin, 1986. doi: 10.1007/BFb0074920.

[20]

R. WangJ. Zhai and T. Zhang, A moderate deviation principle for 2D stochastic Navier-Stokes equations, J. Differ. Equ., 258 (2006), 3363-3390.  doi: 10.1016/j.jde.2015.01.008.

[21]

W. Wang, J. Zhai and T. Zhang, Stochastic 2D Navier-Stokes equations on time-dependent domains, J. Theor. Probab., (2022). doi: 10.1007/s10959-021-01150-0.

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