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A quantum Drift-Diffusion model and its use into a hybrid strategy for strongly confined nanostructures
Time-splitting methods to solve the Hall-MHD systems with Lévy noises
1. | School of Mathematical Sciences, Xiamen University, Xiamen 361005, China |
2. | College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China |
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.
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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