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doi: 10.3934/dcdsb.2022044
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On the global existence to Hall-MHD system

Department of Mathematics and Statistics, Anhui Normal University, Wuhu, 241001, China

Received  January 2021 Revised  January 2022 Early access March 2022

This paper investigates the well-posedness of Hall-magnetohydrodynamics system. By using a new current function $ J = \nabla\times B $ as an additional unknown. The mild solution of Hall-MHD exists globally in the nonhomogeneous Lei-Lin space setting provided that the initial data satisfies $ \|u_{0}\|_{\mathcal{X}^{-1} }+\|B_{0}\|_{ \mathcal{X}^{-1}}+\|J_{0}\|_{ \mathcal{X}^{-1}}<\min \{\frac{\mu }{2}, \frac{\nu }{2}\}. $

Citation: Lvqiao Liu. On the global existence to Hall-MHD system. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022044
References:
[1]

M. AcheritogarayP. DegondA. 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]

H. Bae, Existence and analyticity of Lei-Lin solution to the Navier-Stokes equations, Proc. Amer. Math. Soc., 143 (2015), 2887-2892.  doi: 10.1090/S0002-9939-2015-12266-6.

[3]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, volume 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[4]

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

[5]

M. Cannone and G. Wu, Global well-posedness for Navier-Stokes equations in critical Fourier-Herz spaces, Nonlinear Anal., 75 (2012), 3754-3760.  doi: 10.1016/j.na.2012.01.029.

[6]

D. ChaeP. Degond and J.-G. Liu, Well-posedness for Hall-magnetohydrodynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 555-565.  doi: 10.1016/j.anihpc.2013.04.006.

[7]

D. Chae and J. Lee, On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics, J. Differential Equations, 256 (2014), 3835-3858.  doi: 10.1016/j.jde.2014.03.003.

[8]

J.-Y. Chemin and I. Gallagher, Wellposedness and stability results for the Navier-Stokes equations in ${\bf{R}}^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 599-624.  doi: 10.1016/j.anihpc.2007.05.008.

[9]

A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, volume 4 of Texts in Applied Mathematics, Springer-Verlag, New York, third edition, 1993. doi: 10.1007/978-1-4612-0883-9.

[10]

R. Danchin and J. Tan, On the well-posedness of the Hall-magnetohydrodynamics system in critical spaces, Comm. Partial Differential Equations, 46 (2021), 31-65.  doi: 10.1080/03605302.2020.1822392.

[11]

R. Danchin and J. Tan, The global solvability of the hall-magnetohydrodynamics system in critical sobolev spaces, arXiv e-prints, arXiv: 1912.09194, 2019.

[12]

L. Jlali, Global well posedness of 3D-NSE in Fourier-Lei-Lin spaces, Math. Methods Appl. Sci., 40 (2017), 2713-2736.  doi: 10.1002/mma.4193.

[13]

M. Kwak and B. Lkhagvasuren, Global wellposedness for Hall-MHD equations, Nonlinear Anal., 174 (2018), 104-117.  doi: 10.1016/j.na.2018.04.014.

[14]

Z. Lei and F. Lin, Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math., 64 (2011), 1297-1304.  doi: 10.1002/cpa.20361.

[15]

L. Liu and J. Tan, Global well-posedness for the Hall-magnetohydrodynamics system in larger critical Besov spaces, J. Differential Equations, 274 (2021), 382-413.  doi: 10.1016/j.jde.2020.10.014.

[16]

V. A. Urpin and D. A. Shalybkov, The hall effect and the decay of magnetic fields, Astron. Astrophys., 321 (1997), 685-690. 

[17]

R. Wan and Y. Zhou, On global existence, energy decay and blow-up criteria for the Hall-MHD system, J. Differential Equations, 259 (2015), 5982-6008.  doi: 10.1016/j.jde.2015.07.013.

[18]

R. Wan and Y. Zhou, Global well-posedness, BKM blow-up criteria and zero $h$ limit for the 3D incompressible Hall-MHD equations, J. Differential Equations, 267 (2019), 3724-3747.  doi: 10.1016/j.jde.2019.04.020.

[19]

M. Wardle, Star formation and the hall effect, Astrophys. Space Sci., 292 (2004), 317-323. 

[20]

X. Wu, Y. Yu and Y. Tang, Well-posedness for the incompressible Hall-MHD equations in low regularity spaces, Mediterr. J. Math., 15 (2018), Paper No. 48, 14 pp. doi: 10.1007/s00009-018-1096-x.

[21]

Z. Ye, Global well-posedness and decay results to 3D generalized viscous magnetohydrodynamic equations, Ann. Mat. Pura Appl. (4), 195 (2016), 1111-1121.  doi: 10.1007/s10231-015-0507-x.

show all references

References:
[1]

M. AcheritogarayP. DegondA. 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]

H. Bae, Existence and analyticity of Lei-Lin solution to the Navier-Stokes equations, Proc. Amer. Math. Soc., 143 (2015), 2887-2892.  doi: 10.1090/S0002-9939-2015-12266-6.

[3]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, volume 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[4]

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

[5]

M. Cannone and G. Wu, Global well-posedness for Navier-Stokes equations in critical Fourier-Herz spaces, Nonlinear Anal., 75 (2012), 3754-3760.  doi: 10.1016/j.na.2012.01.029.

[6]

D. ChaeP. Degond and J.-G. Liu, Well-posedness for Hall-magnetohydrodynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 555-565.  doi: 10.1016/j.anihpc.2013.04.006.

[7]

D. Chae and J. Lee, On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics, J. Differential Equations, 256 (2014), 3835-3858.  doi: 10.1016/j.jde.2014.03.003.

[8]

J.-Y. Chemin and I. Gallagher, Wellposedness and stability results for the Navier-Stokes equations in ${\bf{R}}^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 599-624.  doi: 10.1016/j.anihpc.2007.05.008.

[9]

A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, volume 4 of Texts in Applied Mathematics, Springer-Verlag, New York, third edition, 1993. doi: 10.1007/978-1-4612-0883-9.

[10]

R. Danchin and J. Tan, On the well-posedness of the Hall-magnetohydrodynamics system in critical spaces, Comm. Partial Differential Equations, 46 (2021), 31-65.  doi: 10.1080/03605302.2020.1822392.

[11]

R. Danchin and J. Tan, The global solvability of the hall-magnetohydrodynamics system in critical sobolev spaces, arXiv e-prints, arXiv: 1912.09194, 2019.

[12]

L. Jlali, Global well posedness of 3D-NSE in Fourier-Lei-Lin spaces, Math. Methods Appl. Sci., 40 (2017), 2713-2736.  doi: 10.1002/mma.4193.

[13]

M. Kwak and B. Lkhagvasuren, Global wellposedness for Hall-MHD equations, Nonlinear Anal., 174 (2018), 104-117.  doi: 10.1016/j.na.2018.04.014.

[14]

Z. Lei and F. Lin, Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math., 64 (2011), 1297-1304.  doi: 10.1002/cpa.20361.

[15]

L. Liu and J. Tan, Global well-posedness for the Hall-magnetohydrodynamics system in larger critical Besov spaces, J. Differential Equations, 274 (2021), 382-413.  doi: 10.1016/j.jde.2020.10.014.

[16]

V. A. Urpin and D. A. Shalybkov, The hall effect and the decay of magnetic fields, Astron. Astrophys., 321 (1997), 685-690. 

[17]

R. Wan and Y. Zhou, On global existence, energy decay and blow-up criteria for the Hall-MHD system, J. Differential Equations, 259 (2015), 5982-6008.  doi: 10.1016/j.jde.2015.07.013.

[18]

R. Wan and Y. Zhou, Global well-posedness, BKM blow-up criteria and zero $h$ limit for the 3D incompressible Hall-MHD equations, J. Differential Equations, 267 (2019), 3724-3747.  doi: 10.1016/j.jde.2019.04.020.

[19]

M. Wardle, Star formation and the hall effect, Astrophys. Space Sci., 292 (2004), 317-323. 

[20]

X. Wu, Y. Yu and Y. Tang, Well-posedness for the incompressible Hall-MHD equations in low regularity spaces, Mediterr. J. Math., 15 (2018), Paper No. 48, 14 pp. doi: 10.1007/s00009-018-1096-x.

[21]

Z. Ye, Global well-posedness and decay results to 3D generalized viscous magnetohydrodynamic equations, Ann. Mat. Pura Appl. (4), 195 (2016), 1111-1121.  doi: 10.1007/s10231-015-0507-x.

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