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Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise
Gina Cody School of Engineering and Computer Science, 1455 De Maisonneuve Blvd., W. Montreal, QC H3G 1M8, Canada |
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.
References:
[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. 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. |
[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. |
[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. |
[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. 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. |
[12] |
H. Crauel, Random Probability Measures on Polish Spaces, Stochastics Monographs, 11. Taylor & Francis, London, 2002. |
[13] |
H. Crauel, A. 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. 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. |
[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. |
[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. Zabczyk, Stochastic 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. 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. |
[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. 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.
![]() ![]() |
[30] |
G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes, Stochastic Modeling, Chapman & Hall, New York, 1994. |
[31] |
K. Sato, Lévy 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] | |
[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. |
show all references
References:
[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. 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. |
[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. |
[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. |
[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. 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. |
[12] |
H. Crauel, Random Probability Measures on Polish Spaces, Stochastics Monographs, 11. Taylor & Francis, London, 2002. |
[13] |
H. Crauel, A. 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. 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. |
[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. |
[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. Zabczyk, Stochastic 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. 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. |
[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. 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.
![]() ![]() |
[30] |
G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes, Stochastic Modeling, Chapman & Hall, New York, 1994. |
[31] |
K. Sato, Lévy 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] | |
[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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