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September  2018, 38(9): 4745-4765. doi: 10.3934/dcds.2018209

Local and global strong solution to the stochastic 3-D incompressible anisotropic Navier-Stokes equations

School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

* Corresponding author: Ting Zhang

Received  January 2018 Revised  March 2018 Published  June 2018

Fund Project: This work is partially supported by Zhejiang Provincial Natural Science Foundation of China LR17A010001, National Natural Science Foundation of China 11771389, 11331005 and 11621101.

Considering the stochastic 3-D incompressible anisotropic Navier-Stokes equations, we prove the local existence of strong solution in $H^2(\mathbb{T}^3)$. Moreover, we express the probabilistic estimate of the random time interval for the existence of a local solution in terms of expected values of the initial data and the random noise, and establish the global existence of strong solution in probability if the initial data and the random noise are sufficiently small.

Citation: Lihuai Du, Ting Zhang. Local and global strong solution to the stochastic 3-D incompressible anisotropic Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4745-4765. doi: 10.3934/dcds.2018209
References:
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H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343, Springer Science & Business Media, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

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D. BreitE. Feireisl and M. Hofmanová, Local strong solutions to the stochastic compressible Navier-Stokes system, Comm. Partial Differential Equations, 43 (2018), 313-345.  doi: 10.1080/03605302.2018.1442476.  Google Scholar

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M. Capiński and S. Peszat, Local existence and uniqueness of strong solutions to 3-D stochastic Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl, 4 (1997), 185-200.  doi: 10.1007/PL00001415.  Google Scholar

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M. Capiński and S. Peszat, On the existence of a solution to stochastic Navier-Stokes equations, Nonlinear Anal., 44 (2001), 141-177.  doi: 10.1016/S0362-546X(99)00255-2.  Google Scholar

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J.-Y. CheminB. DesjardinsI. Gallagher and E. Grenier, Fluids with anisotropic viscosity, Math. Model. Numer. Anal., 34 (2000), 315-335, Special issue for R.  doi: 10.1051/m2an:2000143.  Google Scholar

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J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Mathematical Geophysics, An Introduction to Rotating Fluids and the Navier-Stokes Equations, vol. 32 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, Oxford, 2006.  Google Scholar

[11]

J.-Y. Chemin and P. Zhang, On the global wellposedness to the 3-D incompressible anisotropic Navier-Stokes equations, Comm. Math. Phys., 272 (2007), 529-566.  doi: 10.1007/s00220-007-0236-0.  Google Scholar

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B. Desjardins and E. Grenier, Derivation of quasi-geostrophic potential vorticity equations, Adv. Differential Equations, 3 (1998), 715-752.   Google Scholar

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F. Flandoli and D. Gątarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.  Google Scholar

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F. Flandoli and B. Maslowski, Ergodicity of the 2-D Navier-Stokes equation under random perturbations, Comm. Math. Phys, 172 (1995), 119-141.  doi: 10.1007/BF02104513.  Google Scholar

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U. Frisch, Turbulence, Cambridge University Press, Cambridge, 1995, The legacy of A. N. Kolmogorov.  Google Scholar

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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

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N. Glatt-Holtz and M. Ziane, Strong pathwise solutions of the stochastic Navier-Stokes system, Adv. Differential Equations, 14 (2009), 567-600.   Google Scholar

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E. Grenier and N. Masmoudi, Ekman layers of rotating fluids, the case of well prepared initial data, Comm. Partial Differential Equations, 22 (1997), 953-975.  doi: 10.1080/03605309708821290.  Google Scholar

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L. Hörmander, The Analysis of Linear Partial Differential Operators. III, vol. 274 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1985, Pseudodifferential operators.  Google Scholar

[23]

D. Iftimie, A uniqueness result for the Navier-Stokes equations with vanishing vertical viscosity, SIAM J. Math. Anal., 33 (2002), 1483-1493.  doi: 10.1137/S0036141000382126.  Google Scholar

[24]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, vol. 113 of Graduate Texts in Mathematics, 2nd edition, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2.  Google Scholar

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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

[26]

J. U. Kim, Existence of a local smooth solution in probability to the stochastic Euler equations in ${\bf{R}}^3$, J. Funct. Anal., 256 (2009), 3660-3687.  doi: 10.1016/j.jfa.2009.03.012.  Google Scholar

[27]

J. U. Kim, Strong Solutions of the Stochastic Navier-Stokes Equations in $\mathbb{R}^3$, Indiana Univ. Math. J., 59 (2010), 1417-1450.  doi: 10.1512/iumj.2010.59.3930.  Google Scholar

[28]

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

[29]

R. Mikulevicius and B. L. Rozovskii, Stochastic Navier-Stokes equations for turbulent flows, SIAM J. Math. Anal, 35 (2004), 1250-1310.  doi: 10.1137/S0036141002409167.  Google Scholar

[30]

R. Mikulevicius and B. L. Rozovskii, Global $L_2$-solutions of stochastic Navier-Stokes equations, Ann. Probab, 33 (2005), 137-176.  doi: 10.1214/009117904000000630.  Google Scholar

[31]

R. Mikulevicius and G. Valiukevicius, On stochastic Euler equation in $\mathbb R^d$, Electron. J. Probab., 5 (2000), 1-20.  doi: 10.1214/EJP.v5-62.  Google Scholar

[32]

M. Paicu, Équation anisotrope de Navier-Stokes dans des espaces critiques, Rev. Mat. Iberoamericana, 21 (2005), 179-235.  doi: 10.4171/RMI/420.  Google Scholar

[33]

M. Paicu, Équation Periodique de Navier-Stokes dans Viscosité une Direction, Comm. Partial Differential Equations, 30 (2005), 1107-1140.  doi: 10.1080/036053005002575529.  Google Scholar

[34]

M. Paicu and P. Zhang, Global solutions to the 3-D incompressible anisotropic Navier-Stokes system in the critical spaces, Comm. Math. Phys., 307 (2011), 713-759.  doi: 10.1007/s00220-011-1350-6.  Google Scholar

[35]

J. Pedlovsky, Geophysical Fluid Dynamics, Springer, Berlin-Heidelberg-New York, 1979. Google Scholar

[36]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[37]

T. Zhang, Global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations in an anisotropic space, Comm. Math. Phys., 287 (2009), 211-224.  doi: 10.1007/s00220-008-0631-1.  Google Scholar

[38]

T. Zhang and D. Y. Fang, Global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations, J. Math. Pures Appl., 90 (2008), 413-449.  doi: 10.1016/j.matpur.2008.06.008.  Google Scholar

show all references

References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343, Springer Science & Business Media, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

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

[3]

H. Bessaih and A. Millet, On stochastic modified 3D Navier-Stokes equations with anisotropic viscosity, J. Math. Anal. Appl., 462 (2018), 915–956, URL https://doi.org/10.1016/j.jmaa.2017.12.053. doi: 10.1016/j.jmaa.2017.12.053.  Google Scholar

[4]

H. Breckner, Galerkin approximation and the strong solution of the Navier-Stokes equation, J. Appl. Math. Stochastic Anal, 13 (1900), 239-259.  doi: 10.1155/S1048953300000228.  Google Scholar

[5]

D. BreitE. Feireisl and M. Hofmanová, Local strong solutions to the stochastic compressible Navier-Stokes system, Comm. Partial Differential Equations, 43 (2018), 313-345.  doi: 10.1080/03605302.2018.1442476.  Google Scholar

[6]

M. Capinski and D. Gatarek, Stochastic equations in Hilbert space with application to Navier-Stokes equations in any dimension, J. Funct. Anal., 126 (1994), 26-35.  doi: 10.1006/jfan.1994.1140.  Google Scholar

[7]

M. Capiński and S. Peszat, Local existence and uniqueness of strong solutions to 3-D stochastic Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl, 4 (1997), 185-200.  doi: 10.1007/PL00001415.  Google Scholar

[8]

M. Capiński and S. Peszat, On the existence of a solution to stochastic Navier-Stokes equations, Nonlinear Anal., 44 (2001), 141-177.  doi: 10.1016/S0362-546X(99)00255-2.  Google Scholar

[9]

J.-Y. CheminB. DesjardinsI. Gallagher and E. Grenier, Fluids with anisotropic viscosity, Math. Model. Numer. Anal., 34 (2000), 315-335, Special issue for R.  doi: 10.1051/m2an:2000143.  Google Scholar

[10]

J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Mathematical Geophysics, An Introduction to Rotating Fluids and the Navier-Stokes Equations, vol. 32 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, Oxford, 2006.  Google Scholar

[11]

J.-Y. Chemin and P. Zhang, On the global wellposedness to the 3-D incompressible anisotropic Navier-Stokes equations, Comm. Math. Phys., 272 (2007), 529-566.  doi: 10.1007/s00220-007-0236-0.  Google Scholar

[12]

G. Da Prato and J. Zabczyk, Stochastic Equations in infinite Dimensions. Second Edition., vol. 152 of Encyclopedia of Mathematics and its Applications, Cambridge university press, Cambrige, 2014. doi: 10.1017/CBO9781107295513.  Google Scholar

[13]

B. Desjardins and E. Grenier, Derivation of quasi-geostrophic potential vorticity equations, Adv. Differential Equations, 3 (1998), 715-752.   Google Scholar

[14]

B. Desjardins and E. Grenier, On the homogeneous model of wind-driven ocean circulation, SIAM J. Appl. Math., 60 (2000), 43-60.  doi: 10.1137/S0036139997324261.  Google Scholar

[15]

L. Du and T. Zhang, Local and global strong solutions to the stochastic incompressible Navier-Stokes equations in critical Besov space, arXiv preprint, arXiv: 1710.11336v2. Google Scholar

[16]

F. Flandoli and D. Gątarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.  Google Scholar

[17]

F. Flandoli and B. Maslowski, Ergodicity of the 2-D Navier-Stokes equation under random perturbations, Comm. Math. Phys, 172 (1995), 119-141.  doi: 10.1007/BF02104513.  Google Scholar

[18]

U. Frisch, Turbulence, Cambridge University Press, Cambridge, 1995, The legacy of A. N. Kolmogorov.  Google Scholar

[19]

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

[20]

N. Glatt-Holtz and M. Ziane, Strong pathwise solutions of the stochastic Navier-Stokes system, Adv. Differential Equations, 14 (2009), 567-600.   Google Scholar

[21]

E. Grenier and N. Masmoudi, Ekman layers of rotating fluids, the case of well prepared initial data, Comm. Partial Differential Equations, 22 (1997), 953-975.  doi: 10.1080/03605309708821290.  Google Scholar

[22]

L. Hörmander, The Analysis of Linear Partial Differential Operators. III, vol. 274 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1985, Pseudodifferential operators.  Google Scholar

[23]

D. Iftimie, A uniqueness result for the Navier-Stokes equations with vanishing vertical viscosity, SIAM J. Math. Anal., 33 (2002), 1483-1493.  doi: 10.1137/S0036141000382126.  Google Scholar

[24]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, vol. 113 of Graduate Texts in Mathematics, 2nd edition, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2.  Google Scholar

[25]

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

[26]

J. U. Kim, Existence of a local smooth solution in probability to the stochastic Euler equations in ${\bf{R}}^3$, J. Funct. Anal., 256 (2009), 3660-3687.  doi: 10.1016/j.jfa.2009.03.012.  Google Scholar

[27]

J. U. Kim, Strong Solutions of the Stochastic Navier-Stokes Equations in $\mathbb{R}^3$, Indiana Univ. Math. J., 59 (2010), 1417-1450.  doi: 10.1512/iumj.2010.59.3930.  Google Scholar

[28]

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

[29]

R. Mikulevicius and B. L. Rozovskii, Stochastic Navier-Stokes equations for turbulent flows, SIAM J. Math. Anal, 35 (2004), 1250-1310.  doi: 10.1137/S0036141002409167.  Google Scholar

[30]

R. Mikulevicius and B. L. Rozovskii, Global $L_2$-solutions of stochastic Navier-Stokes equations, Ann. Probab, 33 (2005), 137-176.  doi: 10.1214/009117904000000630.  Google Scholar

[31]

R. Mikulevicius and G. Valiukevicius, On stochastic Euler equation in $\mathbb R^d$, Electron. J. Probab., 5 (2000), 1-20.  doi: 10.1214/EJP.v5-62.  Google Scholar

[32]

M. Paicu, Équation anisotrope de Navier-Stokes dans des espaces critiques, Rev. Mat. Iberoamericana, 21 (2005), 179-235.  doi: 10.4171/RMI/420.  Google Scholar

[33]

M. Paicu, Équation Periodique de Navier-Stokes dans Viscosité une Direction, Comm. Partial Differential Equations, 30 (2005), 1107-1140.  doi: 10.1080/036053005002575529.  Google Scholar

[34]

M. Paicu and P. Zhang, Global solutions to the 3-D incompressible anisotropic Navier-Stokes system in the critical spaces, Comm. Math. Phys., 307 (2011), 713-759.  doi: 10.1007/s00220-011-1350-6.  Google Scholar

[35]

J. Pedlovsky, Geophysical Fluid Dynamics, Springer, Berlin-Heidelberg-New York, 1979. Google Scholar

[36]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[37]

T. Zhang, Global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations in an anisotropic space, Comm. Math. Phys., 287 (2009), 211-224.  doi: 10.1007/s00220-008-0631-1.  Google Scholar

[38]

T. Zhang and D. Y. Fang, Global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations, J. Math. Pures Appl., 90 (2008), 413-449.  doi: 10.1016/j.matpur.2008.06.008.  Google Scholar

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