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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 and 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.

[2]

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

[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.

[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. 

[5]

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

[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.

[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.

[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.

[9]

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

[10]

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

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[22]

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

[23]

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

[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.

[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.

[26]

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

[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.

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.

[2]

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

[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.

[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. 

[5]

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

[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.

[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.

[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.

[9]

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

[10]

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

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[22]

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

[23]

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

[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.

[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.

[26]

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

[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.

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