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October  2021, 26(10): 5421-5448. doi: 10.3934/dcdsb.2020352

Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise

Leanne Dong ,

Gina Cody School of Engineering and Computer Science, 1455 De Maisonneuve Blvd., W. Montreal, QC H3G 1M8, Canada

Corresponding author: Leanne Dong

Received  November 2019 Revised  June 2020 Published  October 2021 Early access  December 2020

In this paper we prove that the stochastic Navier-Stokes equations with stable Lévy noise generate a random dynamical systems. Then we prove the existence of random attractor for the Navier-Stokes equations on 2D spheres under stable Lévy noise (finite dimensional). We also deduce the existence of a Feller Markov Invariant Measure.

Keywords: Random attractors, random dynamical systems, stochastic Navier-Stokes, unit spheres, stable Lévy noise, Feller Markov invariant measure.
Mathematics Subject Classification: Primary: 60H15, 35R60; Secondary: 37H10, 34F05.
Citation: Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5421-5448. doi: 10.3934/dcdsb.2020352
References:
[1]

D. Applebaum, Lévy processes and stochastic integrals in Banach spaces, Probab. Math. Statist., 27 (2007), 75-88.   Google Scholar

[2]

L. Arnold, Random Dynamical Systems, Springer Science & Business Media, 2013. Google Scholar

[3]

J.-P. Bouchaud and A. George, Anomalous diffusion in disordered media: Statistic mechanics, models and physical applications, Phys. Rep, 195 (1990), 127-293.  doi: 10.1016/0370-1573(90)90099-N.  Google Scholar

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[5]

Z. Brzeźniak, M. Capiński and F. Flandoli, Pathwise global attractors for stationary random dynamical systems, Probab. Theory Related Fields, 95 (1993), 87-102.  doi: 10.1007/BF01197339.  Google Scholar

[6]

Z. Brzeźiak, B. Goldys and Q. T. Le Gia, Random dynamical systems generated by stochastic Navier-Stokes equations on a rotating sphere, J. Math. Anal. Appl., 426 (2015), 505-545.  doi: 10.1016/j.jmaa.2015.01.054.  Google Scholar

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Z. Brzeźiak, B. Goldys and Q. T. Le Gia, Random attractors for the stochastic Navier–Stokes equations on the 2D unit sphere, J. Math. Fluid Mech., 20 (2018), 227-253.  doi: 10.1007/s00021-017-0351-4.  Google Scholar

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Z. Brzeźiak and Y. Li, Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains, Trans. Amer. Math. Soc., 358 (2006), 5587-5629.  doi: 10.1090/S0002-9947-06-03923-7.  Google Scholar

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Z. Brzeźiak and J. Zabczyk, Regularity of Ornstein-Uhlenbeck processes driven by a Lévy white noise, Potential Anal., 32 (2010), 153-188.  doi: 10.1007/s11118-009-9149-1.  Google Scholar

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C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol. 580. Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[11]

M. D. Chekroun, E. Simonnet and M. Ghil, Stochastic climate dynamics: Random attractors and time-dependent invariant measures, Phys. D, 240 (2011), 1685-1700.  doi: 10.1016/j.physd.2011.06.005.  Google Scholar

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H. Crauel, Random Probability Measures on Polish Spaces, Stochastics Monographs, 11. Taylor & Francis, London, 2002.  Google Scholar

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H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar

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H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

[15]

L. Dong, Invariant measures for the stochastic navier-stokes equation on a 2D rotating sphere with stable Lévy noise, arXiv e-prints, arXiv: 1812.05513. Google Scholar

[16]

L. Dong, Strong solutions for the stochastic Navier-Stokes equations on the 2D rotating sphere with stable Lévy noise, J. Math. Anal. Appl., 489 (2020), 124182, 37 pp. doi: 10.1016/j.jmaa.2020.124182.  Google Scholar

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S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, John Wiley & Sons, Inc., New York, 1986. doi: 10.1002/9780470316658.  Google Scholar

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T. Gao, J. Duan and X. Li, Fokker-Planck equations for stochastic dynamical systems with symmetric Lévy motions, Appl. Math. Comput., 278 (2016), 1-20.  doi: 10.1016/j.amc.2016.01.010.  Google Scholar

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B. Gess, W. Liu and M. Röckner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, J. Differential Equations, 251 (2011), 1225-1253.  doi: 10.1016/j.jde.2011.02.013.  Google Scholar

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G. A. Gottwald and D. T. Crommelin and C. L. E. Franzke, Stochastic climate theory, Nonlinear and Stochastic Climate Dynamics, Cambridge Univ. Press, Cambridge, (2017), 209–240.  Google Scholar

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A. Gu, Synchronization of coupled stochastic systems driven by $\alpha$-stable Lévy noises, Math. Probl. Eng., 2013 (2013), Art. ID 685798, 10 pp. doi: 10.1155/2013/685798.  Google Scholar

[22]

A. Gu and W. Ai, Random attractor for stochastic lattice dynamical systems with $\alpha$-stable Lévy noises, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 1433-1441.  doi: 10.1016/j.cnsns.2013.08.036.  Google Scholar

[23]

J. Huang, Y. Li and J. Duan, Random dynamics of the stochastic Boussinesq equations driven by Lévy noises, Abstr. Appl. Anal., 2013 (2013), Art. ID 653160, 10 pp. doi: 10.1155/2013/653160.  Google Scholar

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M. Ledous and M. Talagrand, Probability in Banach Spaces: Isoperimetry and Processes, Springer Science & Business Media, 2013. Google Scholar

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F. Matthäus, M. S. Mommer, T. Curk and J. Dobnikar, On the origin and characteristics of noise-induced Lévy walks of E. Coli, PLoS ONE, 6 (2011), e18623. doi: 10.1371/journal.pone.0018623.  Google Scholar

[26] S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations With Lévy Noise, Encyclopedia of Mathematics and its Applications, 113. Cambridge University Press, Cambridge, 2007.  doi: 10.1017/CBO9780511721373.  Google Scholar
[27]

G. W. Peters, S. A. Sisson and Y. Fan, Likelihood-free Bayesian inference for $\alpha$-stable models, Comput. Statist. Data Anal., 56 (2012), 3743-3756.  doi: 10.1016/j.csda.2010.10.004.  Google Scholar

[28]

E. Priola and J. Zabczyk, Structural properties of semilinear SPDEs driven by cylindrical stable processe, Probab. Theory Related Fields, 149 (2011), 97-137.  doi: 10.1007/s00440-009-0243-5.  Google Scholar

[29] J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2001.   Google Scholar
[30]

G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes, Stochastic Modeling, Chapman & Hall, New York, 1994.  Google Scholar

[31] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999.   Google Scholar
[32]

L. Serdukova, Y. Zheng, J. Duan and J. Kurths, Metastability for discontinuous dynamical systems under Lévy noise: Case study on Amazonian Vegetation, Scientific Reports, 7 (2017), Article number, 9336. doi: 10.1038/s41598-017-07686-8.  Google Scholar

[33]

M. Talagrand, Upper and Lower Bounds for Stochastic Processes: Modern Methods and Classical Problems, Springer, Heidelberg, 2014. doi: 10.1007/978-3-642-54075-2.  Google Scholar

[34]

R. Temam, Infinite-Dimensional Dynamical Systems, 1993. Google Scholar

[35]

L. Xu, Applications of a simple but useful technique to stochastic convolution of $\alpha$-stable processes, arXiv e-prints, arXiv: 1201.4260. Google Scholar

[36]

F. Yonezawa, Introduction to focused session on'anomalous relaxation', Journal of Non-Cryst. Solids, 198-200 (1996), 503-506.  doi: 10.1016/0022-3093(95)00726-1.  Google Scholar

[37]

Y. Zhang, Z. Cheng, X. Zhang, X. Chen, J. Duan and X. Li, Data assimilation and parameter estimation for a multiscale stochastic system with $\alpha$-stable Lévy noise, J. Stat. Mech. Theory Exp., 11 (2017), 113401, 17 pp. doi: 10.1088/1742-5468/aa9343.  Google Scholar

show all references

References:
[1]

D. Applebaum, Lévy processes and stochastic integrals in Banach spaces, Probab. Math. Statist., 27 (2007), 75-88.   Google Scholar

[2]

L. Arnold, Random Dynamical Systems, Springer Science & Business Media, 2013. Google Scholar

[3]

J.-P. Bouchaud and A. George, Anomalous diffusion in disordered media: Statistic mechanics, models and physical applications, Phys. Rep, 195 (1990), 127-293.  doi: 10.1016/0370-1573(90)90099-N.  Google Scholar

[4]

Z. Brzeźiak, Asymptotic compactness and absorbing sets for stochastic Burgers' equations driven by space-time white noise and for some two-dimensional stochastic Navier-Stokes equations on certain unbounded domains, Stochastic Partial Differential Equations and Applications–VII, Lect. Notes Pure Appl. Math., Chapman & Hall/CRC, Boca Raton, FL, 245 (2006), 35–52.  Google Scholar

[5]

Z. Brzeźniak, M. Capiński and F. Flandoli, Pathwise global attractors for stationary random dynamical systems, Probab. Theory Related Fields, 95 (1993), 87-102.  doi: 10.1007/BF01197339.  Google Scholar

[6]

Z. Brzeźiak, B. Goldys and Q. T. Le Gia, Random dynamical systems generated by stochastic Navier-Stokes equations on a rotating sphere, J. Math. Anal. Appl., 426 (2015), 505-545.  doi: 10.1016/j.jmaa.2015.01.054.  Google Scholar

[7]

Z. Brzeźiak, B. Goldys and Q. T. Le Gia, Random attractors for the stochastic Navier–Stokes equations on the 2D unit sphere, J. Math. Fluid Mech., 20 (2018), 227-253.  doi: 10.1007/s00021-017-0351-4.  Google Scholar

[8]

Z. Brzeźiak and Y. Li, Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains, Trans. Amer. Math. Soc., 358 (2006), 5587-5629.  doi: 10.1090/S0002-9947-06-03923-7.  Google Scholar

[9]

Z. Brzeźiak and J. Zabczyk, Regularity of Ornstein-Uhlenbeck processes driven by a Lévy white noise, Potential Anal., 32 (2010), 153-188.  doi: 10.1007/s11118-009-9149-1.  Google Scholar

[10]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol. 580. Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[11]

M. D. Chekroun, E. Simonnet and M. Ghil, Stochastic climate dynamics: Random attractors and time-dependent invariant measures, Phys. D, 240 (2011), 1685-1700.  doi: 10.1016/j.physd.2011.06.005.  Google Scholar

[12]

H. Crauel, Random Probability Measures on Polish Spaces, Stochastics Monographs, 11. Taylor & Francis, London, 2002.  Google Scholar

[13]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar

[14]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

[15]

L. Dong, Invariant measures for the stochastic navier-stokes equation on a 2D rotating sphere with stable Lévy noise, arXiv e-prints, arXiv: 1812.05513. Google Scholar

[16]

L. Dong, Strong solutions for the stochastic Navier-Stokes equations on the 2D rotating sphere with stable Lévy noise, J. Math. Anal. Appl., 489 (2020), 124182, 37 pp. doi: 10.1016/j.jmaa.2020.124182.  Google Scholar

[17]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, John Wiley & Sons, Inc., New York, 1986. doi: 10.1002/9780470316658.  Google Scholar

[18]

T. Gao, J. Duan and X. Li, Fokker-Planck equations for stochastic dynamical systems with symmetric Lévy motions, Appl. Math. Comput., 278 (2016), 1-20.  doi: 10.1016/j.amc.2016.01.010.  Google Scholar

[19]

B. Gess, W. Liu and M. Röckner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, J. Differential Equations, 251 (2011), 1225-1253.  doi: 10.1016/j.jde.2011.02.013.  Google Scholar

[20]

G. A. Gottwald and D. T. Crommelin and C. L. E. Franzke, Stochastic climate theory, Nonlinear and Stochastic Climate Dynamics, Cambridge Univ. Press, Cambridge, (2017), 209–240.  Google Scholar

[21]

A. Gu, Synchronization of coupled stochastic systems driven by $\alpha$-stable Lévy noises, Math. Probl. Eng., 2013 (2013), Art. ID 685798, 10 pp. doi: 10.1155/2013/685798.  Google Scholar

[22]

A. Gu and W. Ai, Random attractor for stochastic lattice dynamical systems with $\alpha$-stable Lévy noises, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 1433-1441.  doi: 10.1016/j.cnsns.2013.08.036.  Google Scholar

[23]

J. Huang, Y. Li and J. Duan, Random dynamics of the stochastic Boussinesq equations driven by Lévy noises, Abstr. Appl. Anal., 2013 (2013), Art. ID 653160, 10 pp. doi: 10.1155/2013/653160.  Google Scholar

[24]

M. Ledous and M. Talagrand, Probability in Banach Spaces: Isoperimetry and Processes, Springer Science & Business Media, 2013. Google Scholar

[25]

F. Matthäus, M. S. Mommer, T. Curk and J. Dobnikar, On the origin and characteristics of noise-induced Lévy walks of E. Coli, PLoS ONE, 6 (2011), e18623. doi: 10.1371/journal.pone.0018623.  Google Scholar

[26] S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations With Lévy Noise, Encyclopedia of Mathematics and its Applications, 113. Cambridge University Press, Cambridge, 2007.  doi: 10.1017/CBO9780511721373.  Google Scholar
[27]

G. W. Peters, S. A. Sisson and Y. Fan, Likelihood-free Bayesian inference for $\alpha$-stable models, Comput. Statist. Data Anal., 56 (2012), 3743-3756.  doi: 10.1016/j.csda.2010.10.004.  Google Scholar

[28]

E. Priola and J. Zabczyk, Structural properties of semilinear SPDEs driven by cylindrical stable processe, Probab. Theory Related Fields, 149 (2011), 97-137.  doi: 10.1007/s00440-009-0243-5.  Google Scholar

[29] J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2001.   Google Scholar
[30]

G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes, Stochastic Modeling, Chapman & Hall, New York, 1994.  Google Scholar

[31] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999.   Google Scholar
[32]

L. Serdukova, Y. Zheng, J. Duan and J. Kurths, Metastability for discontinuous dynamical systems under Lévy noise: Case study on Amazonian Vegetation, Scientific Reports, 7 (2017), Article number, 9336. doi: 10.1038/s41598-017-07686-8.  Google Scholar

[33]

M. Talagrand, Upper and Lower Bounds for Stochastic Processes: Modern Methods and Classical Problems, Springer, Heidelberg, 2014. doi: 10.1007/978-3-642-54075-2.  Google Scholar

[34]

R. Temam, Infinite-Dimensional Dynamical Systems, 1993. Google Scholar

[35]

L. Xu, Applications of a simple but useful technique to stochastic convolution of $\alpha$-stable processes, arXiv e-prints, arXiv: 1201.4260. Google Scholar

[36]

F. Yonezawa, Introduction to focused session on'anomalous relaxation', Journal of Non-Cryst. Solids, 198-200 (1996), 503-506.  doi: 10.1016/0022-3093(95)00726-1.  Google Scholar

[37]

Y. Zhang, Z. Cheng, X. Zhang, X. Chen, J. Duan and X. Li, Data assimilation and parameter estimation for a multiscale stochastic system with $\alpha$-stable Lévy noise, J. Stat. Mech. Theory Exp., 11 (2017), 113401, 17 pp. doi: 10.1088/1742-5468/aa9343.  Google Scholar

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