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

  • Corresponding author: Leanne Dong

    Corresponding author: Leanne Dong
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  • 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.

    Mathematics Subject Classification: Primary: 60H15, 35R60; Secondary: 37H10, 34F05.


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  • [1] D. Applebaum, Lévy processes and stochastic integrals in Banach spaces, Probab. Math. Statist., 27 (2007), 75-88. 
    [2] L. Arnold, Random Dynamical Systems, Springer Science & Business Media, 2013.
    [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.
    [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.
    [5] Z. BrzeźniakM. Capiński and F. Flandoli, Pathwise global attractors for stationary random dynamical systems, Probab. Theory Related Fields, 95 (1993), 87-102.  doi: 10.1007/BF01197339.
    [6] Z. BrzeźiakB. 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.
    [7] Z. BrzeźiakB. 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.
    [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.
    [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.
    [10] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol. 580. Springer-Verlag, Berlin-New York, 1977.
    [11] M. D. ChekrounE. 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.
    [12] H. Crauel, Random Probability Measures on Polish Spaces, Stochastics Monographs, 11. Taylor & Francis, London, 2002.
    [13] H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.
    [14] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.
    [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.
    [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.
    [17] S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, John Wiley & Sons, Inc., New York, 1986. doi: 10.1002/9780470316658.
    [18] T. GaoJ. 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.
    [19] B. GessW. 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.
    [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.
    [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.
    [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.
    [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.
    [24] M. Ledous and M. Talagrand, Probability in Banach Spaces: Isoperimetry and Processes, Springer Science & Business Media, 2013.
    [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.
    [26] S. Peszat and  J. ZabczykStochastic Partial Differential Equations With Lévy Noise, Encyclopedia of Mathematics and its Applications, 113. Cambridge University Press, Cambridge, 2007.  doi: 10.1017/CBO9780511721373.
    [27] G. W. PetersS. 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.
    [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.
    [29] J. C. RobinsonInfinite-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. 
    [30] G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes, Stochastic Modeling, Chapman & Hall, New York, 1994.
    [31] K. SatoLévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999. 
    [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.
    [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.
    [34] R. Temam, Infinite-Dimensional Dynamical Systems, 1993.
    [35] L. Xu, Applications of a simple but useful technique to stochastic convolution of $\alpha$-stable processes, arXiv e-prints, arXiv: 1201.4260.
    [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.
    [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.
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