American Institute of Mathematical Sciences

Advanced
February  2019, 12(1): 243-267. doi: 10.3934/krm.2019011

Time-splitting methods to solve the Hall-MHD systems with Lévy noises

Zhong Tan 1, , Huaqiao Wang 1,2, and  Yucong Wang 1,2,,

1. 

School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
2. 

College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China

* Corresponding author: Yucong Wang

Received  March 2017 Revised  March 2018 Published  July 2018

Fund Project: Z. Tan and Y.C. Wang is supported by the National Natural Science Foundation of China No. 11271305, 11531010. H. Wang is supported by National Postdoctoral Program for Innovative Talents No. BX201600020.

In this paper, we establish the existence of a martingale solution to the stochastic incompressible Hall-MHD systems with Lévy noises in a bounded domain. The proof is based on a new method, i.e., the time splitting method and the stochastic compactness method.

Keywords: Hall-MHD systems, Lévy noises, martingale solutions, stochastic compactness method, time splitting method.
Mathematics Subject Classification: Primary: 35Q35, 60H15, 76N10.
Citation: Zhong Tan, Huaqiao Wang, Yucong Wang. Time-splitting methods to solve the Hall-MHD systems with Lévy noises. Kinetic & Related Models, 2019, 12 (1) : 243-267. doi: 10.3934/krm.2019011
References:
[1]

M. Acheritogary, P. Degond, A. Frouvelle and J.-G. Liu, Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system, Kinet. Relat. Models., 4 (2011), 901-918.  doi: 10.3934/krm.2011.4.901.  Google Scholar

[2]

D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511755323.  Google Scholar

[3]

V. Barbu and G. Da Prato, Existence and ergodicity for the 2D stochastic MHD equations, Appl. Math. Optim., 56 (2007), 145-168.  doi: 10.1007/s00245-007-0882-2.  Google Scholar

[4]

M. J. Benvenutti and L. C. F. Ferreira, Existence and stability of global large strong solutions for the Hall-MHD system, Mathematics, 29 (2016), 977-1000.   Google Scholar

[5]

F. Berthelin and J. Vovelle, Stochastic isentropic Euler equations, Mathematics, (2013), 1-54.   Google Scholar

[6]

Z. Brzeźniak and E. Hausenblas, Uniqueness of the Stochastic Integral Driven by Lévy Processes, in: Seminar on Stochastic Analysis, Random Fields and Applications VI, Birkhäuser, 2011. Google Scholar

[7]

Z. Brzeźniak and E. Hausenblas, Martingale solutions for stochastic equations of reaction diffusion type driven by Lévy noise or Poisson random measure, Preprint, arXiv: math/1010.5933v1. Google Scholar

[8]

I. Chueshov and A. Millet, Stochastic 2D hydrodynamical type systems: Well posedeness and large deviations, Applied Mathematics & Optimization, 61 (2010), 379-420.  doi: 10.1007/s00245-009-9091-z.  Google Scholar

[9]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.  Google Scholar

[10]

T. G. Forbes, Magnetic reconnection in solar flares, Geophysical Fluid Dynamics., 62 (1991), 15-36.  doi: 10.1080/03091929108229123.  Google Scholar

[11]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, 1981.  Google Scholar

[12]

A. Jakubowski, The a.s. Skorokhod representation for subsequences in nonmetric spaces, Teor. Veroyatnost. i Primenen., 42 (1997), 209-216.  doi: 10.4213/tvp1769.  Google Scholar

[13]

A. Joffe and M. Métivier, Weak convergence of sequences of semimartingales with applications to multitype branching processes, Adv. Appl. Prob., 18 (1986), 20-65.  doi: 10.2307/1427238.  Google Scholar

[14]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2.  Google Scholar

[15]

M. J. Lighthill, Studies on magneto-hydrodynamics waves and other anisogtropic wave motion, Philo. Trans. R. Soc. Lond. Ser A., 252 (1960), 397-430.  doi: 10.1098/rsta.1960.0010.  Google Scholar

[16]

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.  Google Scholar

[17]

P. D. Mininni, D. O. Gomez and S. M. Mahajan, Dynamo Action in magnetohydrodynamics and Hall magnetohydrodynamics, Astrophys. J., 587 (2003), 472-481.  doi: 10.1086/368181.  Google Scholar

[18]

E. Motyl, Stochastic Navier-Stokes Equations driven by Levy noise in unbounded 3D domains, Potential Anal., 38 (2013), 863-912.  doi: 10.1007/s11118-012-9300-2.  Google Scholar

[19]

E. Motyl, Stochastic hydrodynamic-type evolution equations driven by Lévy noise in 3D unbounded domains-Abstract framework and applications, Stochasitc Process. Appl., 124 (2014), 2052-2097.  doi: 10.1016/j.spa.2014.01.009.  Google Scholar

[20]

S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511721373.  Google Scholar

[21]

M. Sango, Magnetohydrodynamic turbulent flows: Existence results, Phys. D., 239 (2010), 912-923.  doi: 10.1016/j.physd.2010.01.009.  Google Scholar

[22]

K. I. Sato, Lévy Processes and Infinite Divisible Distributions, Cambridge University Press, Cambridge, 1999.  Google Scholar

[23]

D. A. Shalybkov and V. A. Urpin, The Hall effect and the decay of magnetic fields, Astronomy & Astrophysics., 321 (1997), 685-690.   Google Scholar

[24]

A. N. Simakov and L. Chacón, Quantitative, analytical model for magnetic reconnection in Hall magnetohydrodynamics, Physics of Plasmas, 16 (2009), 055701.  doi: 10.1063/1.3077269.  Google Scholar

[25]

S.S. Sritharan and P. Sundar, The stochastic magneto-hydrodynamic system, Infinite Dimensional Analysis Quantum Probability & Related Topics, 2 (1999), 241-265.  doi: 10.1142/S0219025799000138.  Google Scholar

[26]

P. Sundar, Stochastic magnetohydrodynamic system perturbed by general noise, Commun. Stoch. Anal., 4 (2010), 253-269.   Google Scholar

[27]

Z. Tan, D. Wang and H. Wang, Global strong solution to the three-dimensional stochastic incompressible magnetohydrodynamic equations, Math. Ann., 365 (2016), 1219-1256.  doi: 10.1007/s00208-015-1296-7.  Google Scholar

show all references

References:
[1]

M. Acheritogary, P. Degond, A. Frouvelle and J.-G. Liu, Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system, Kinet. Relat. Models., 4 (2011), 901-918.  doi: 10.3934/krm.2011.4.901.  Google Scholar

[2]

D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511755323.  Google Scholar

[3]

V. Barbu and G. Da Prato, Existence and ergodicity for the 2D stochastic MHD equations, Appl. Math. Optim., 56 (2007), 145-168.  doi: 10.1007/s00245-007-0882-2.  Google Scholar

[4]

M. J. Benvenutti and L. C. F. Ferreira, Existence and stability of global large strong solutions for the Hall-MHD system, Mathematics, 29 (2016), 977-1000.   Google Scholar

[5]

F. Berthelin and J. Vovelle, Stochastic isentropic Euler equations, Mathematics, (2013), 1-54.   Google Scholar

[6]

Z. Brzeźniak and E. Hausenblas, Uniqueness of the Stochastic Integral Driven by Lévy Processes, in: Seminar on Stochastic Analysis, Random Fields and Applications VI, Birkhäuser, 2011. Google Scholar

[7]

Z. Brzeźniak and E. Hausenblas, Martingale solutions for stochastic equations of reaction diffusion type driven by Lévy noise or Poisson random measure, Preprint, arXiv: math/1010.5933v1. Google Scholar

[8]

I. Chueshov and A. Millet, Stochastic 2D hydrodynamical type systems: Well posedeness and large deviations, Applied Mathematics & Optimization, 61 (2010), 379-420.  doi: 10.1007/s00245-009-9091-z.  Google Scholar

[9]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.  Google Scholar

[10]

T. G. Forbes, Magnetic reconnection in solar flares, Geophysical Fluid Dynamics., 62 (1991), 15-36.  doi: 10.1080/03091929108229123.  Google Scholar

[11]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, 1981.  Google Scholar

[12]

A. Jakubowski, The a.s. Skorokhod representation for subsequences in nonmetric spaces, Teor. Veroyatnost. i Primenen., 42 (1997), 209-216.  doi: 10.4213/tvp1769.  Google Scholar

[13]

A. Joffe and M. Métivier, Weak convergence of sequences of semimartingales with applications to multitype branching processes, Adv. Appl. Prob., 18 (1986), 20-65.  doi: 10.2307/1427238.  Google Scholar

[14]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2.  Google Scholar

[15]

M. J. Lighthill, Studies on magneto-hydrodynamics waves and other anisogtropic wave motion, Philo. Trans. R. Soc. Lond. Ser A., 252 (1960), 397-430.  doi: 10.1098/rsta.1960.0010.  Google Scholar

[16]

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.  Google Scholar

[17]

P. D. Mininni, D. O. Gomez and S. M. Mahajan, Dynamo Action in magnetohydrodynamics and Hall magnetohydrodynamics, Astrophys. J., 587 (2003), 472-481.  doi: 10.1086/368181.  Google Scholar

[18]

E. Motyl, Stochastic Navier-Stokes Equations driven by Levy noise in unbounded 3D domains, Potential Anal., 38 (2013), 863-912.  doi: 10.1007/s11118-012-9300-2.  Google Scholar

[19]

E. Motyl, Stochastic hydrodynamic-type evolution equations driven by Lévy noise in 3D unbounded domains-Abstract framework and applications, Stochasitc Process. Appl., 124 (2014), 2052-2097.  doi: 10.1016/j.spa.2014.01.009.  Google Scholar

[20]

S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511721373.  Google Scholar

[21]

M. Sango, Magnetohydrodynamic turbulent flows: Existence results, Phys. D., 239 (2010), 912-923.  doi: 10.1016/j.physd.2010.01.009.  Google Scholar

[22]

K. I. Sato, Lévy Processes and Infinite Divisible Distributions, Cambridge University Press, Cambridge, 1999.  Google Scholar

[23]

D. A. Shalybkov and V. A. Urpin, The Hall effect and the decay of magnetic fields, Astronomy & Astrophysics., 321 (1997), 685-690.   Google Scholar

[24]

A. N. Simakov and L. Chacón, Quantitative, analytical model for magnetic reconnection in Hall magnetohydrodynamics, Physics of Plasmas, 16 (2009), 055701.  doi: 10.1063/1.3077269.  Google Scholar

[25]

S.S. Sritharan and P. Sundar, The stochastic magneto-hydrodynamic system, Infinite Dimensional Analysis Quantum Probability & Related Topics, 2 (1999), 241-265.  doi: 10.1142/S0219025799000138.  Google Scholar

[26]

P. Sundar, Stochastic magnetohydrodynamic system perturbed by general noise, Commun. Stoch. Anal., 4 (2010), 253-269.   Google Scholar

[27]

Z. Tan, D. Wang and H. Wang, Global strong solution to the three-dimensional stochastic incompressible magnetohydrodynamic equations, Math. Ann., 365 (2016), 1219-1256.  doi: 10.1007/s00208-015-1296-7.  Google Scholar

[1]

Kumarasamy Sakthivel, Sivaguru S. Sritharan. Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise. Evolution Equations & Control Theory, 2012, 1 (2) : 355-392. doi: 10.3934/eect.2012.1.355
[2]

Ning Duan, Yasuhide Fukumoto, Xiaopeng Zhao. Asymptotic behavior of solutions to incompressible electron inertial Hall-MHD system in $ \mathbb{R}^3 $. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3035-3057. doi: 10.3934/cpaa.2019136
[3]

Jincheng Gao, Zheng-An Yao. Global existence and optimal decay rates of solutions for compressible Hall-MHD equations. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3077-3106. doi: 10.3934/dcds.2016.36.3077
[4]

Min Niu, Bin Xie. Comparison theorem and correlation for stochastic heat equations driven by Lévy space-time white noises. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 2989-3009. doi: 10.3934/dcdsb.2018296
[5]

Jishan Fan, Fucai Li, Gen Nakamura. Low Mach number limit of the full compressible Hall-MHD system. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1731-1740. doi: 10.3934/cpaa.2017084
[6]

Hussain Alazki, Alexander Poznyak. Robust output stabilization for a class of nonlinear uncertain stochastic systems under multiplicative and additive noises: The attractive ellipsoid method. Journal of Industrial & Management Optimization, 2016, 12 (1) : 169-186. doi: 10.3934/jimo.2016.12.169
[7]

Jiahui Zhu, Zdzisław Brzeźniak. Nonlinear stochastic partial differential equations of hyperbolic type driven by Lévy-type noises. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3269-3299. doi: 10.3934/dcdsb.2016097
[8]

Jiangtao Yang. Permanence, extinction and periodic solution of a stochastic single-species model with Lévy noises. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5641-5660. doi: 10.3934/dcdsb.2020371
[9]

Liejune Shiau, Roland Glowinski. Operator splitting method for friction constrained dynamical systems. Conference Publications, 2005, 2005 (Special) : 806-815. doi: 10.3934/proc.2005.2005.806
[10]

Yulin Song. Density functions of distribution dependent SDEs driven by Lévy noises. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2399-2419. doi: 10.3934/cpaa.2021087
[11]

Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209
[12]

Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221
[13]

Wen Chen, Song Wang. A finite difference method for pricing European and American options under a geometric Lévy process. Journal of Industrial & Management Optimization, 2015, 11 (1) : 241-264. doi: 10.3934/jimo.2015.11.241
[14]

Danielle Hilhorst, Masato Iida, Masayasu Mimura, Hirokazu Ninomiya. Relative compactness in $L^p$ of solutions of some 2m components competition-diffusion systems. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 233-244. doi: 10.3934/dcds.2008.21.233
[15]

Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133
[16]

Badr-eddine Berrhazi, Mohamed El Fatini, Tomás Caraballo, Roger Pettersson. A stochastic SIRI epidemic model with Lévy noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2415-2431. doi: 10.3934/dcdsb.2018057
[17]

G. Deugoué, B. Jidjou Moghomye, T. Tachim Medjo. Approximation of a stochastic two-phase flow model by a splitting-up method. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1135-1170. doi: 10.3934/cpaa.2021010
[18]

Nathan Glatt-Holtz, Roger Temam, Chuntian Wang. Martingale and pathwise solutions to the stochastic Zakharov-Kuznetsov equation with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1047-1085. doi: 10.3934/dcdsb.2014.19.1047
[19]

Gengen Zhang. Time splitting combined with exponential wave integrator Fourier pseudospectral method for quantum Zakharov system. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021149
[20]

Manil T. Mohan, Sivaguru S. Sritharan. $\mathbb{L}^p-$solutions of the stochastic Navier-Stokes equations subject to Lévy noise with $\mathbb{L}^m(\mathbb{R}^m)$ initial data. Evolution Equations & Control Theory, 2017, 6 (3) : 409-425. doi: 10.3934/eect.2017021

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (273)
  • HTML views (276)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy