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Global well-posedness for the Hall-magnetohydrodynamics system in larger critical Besov spaces. (English) Zbl 1454.35291

Summary: We prove the global well-posedness of the Cauchy problem to the 3D incompressible Hall-magnetohydrodynamic system supplemented with initial data in critical Besov spaces that allowing for different integrability indices for the velocity field \(u\) and magnetic field \(b\) (and its current \(J)\), which generalize the result in [R. Danchin and the second author, “On the well-posedness of the Hall-magnetohydrodynamics system in critical spaces”, Commun. Partial Differ. Equations 46, No. 1, 31–65 (2020; doi:10.1080/03605302.2020.18223)]. Meanwhile, we analyze the long-time behavior of the solutions and get some decay estimates. Finally, a stability theorem for global solutions is established.

MSC:

35Q35 PDEs in connection with fluid mechanics
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76W05 Magnetohydrodynamics and electrohydrodynamics
35B35 Stability in context of PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
86A10 Meteorology and atmospheric physics
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References:

[1] Acheritogaray, M.; Degond, P.; Frouvelle, A.; Liu, J.-G., Kinetic formulation and global existence for the Hall-magneto-hydrodynamics system, Kinet. Relat. Models, 4, 901-918 (2011) · Zbl 1251.35076
[2] Bahouri, H.; Chemin, J.-Y.; Danchin, R., Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, vol. 343 (2011), Springer: Springer Heidelberg · Zbl 1227.35004
[3] Balbus, S. A.; Terquem, C., Linear analysis of the Hall effect in protostellar disks, Astrophys. J., 552, 235-247 (2001)
[4] Benvenutti, M. J.; Ferreira, L. C.F., Existence and stability of global large strong solutions for the Hall-MHD system, Differ. Integral Equ., 29, 977-1000 (2016) · Zbl 1389.35255
[5] Bony, J.-M., Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. Éc. Norm. Supér. (4), 14, 209-246 (1981) · Zbl 0495.35024
[6] C.R. Braiding, M. Wardle, Star formation and the Hall effect, 2005.
[7] Chae, D.; Degond, P.; Liu, J.-G., Well-posedness for Hall-magnetohydrodynamics, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 31, 555-565 (2014) · Zbl 1297.35064
[8] Chae, D.; Lee, J., On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics, J. Differ. Equ., 256, 3835-3858 (2014) · Zbl 1295.35122
[9] Chae, D.; Weng, S., Singularity formation for the incompressible Hall-MHD equations without resistivity, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 33, 4, 1009-1022 (2016) · Zbl 1347.35199
[10] Chemin, J.-Y., Remarques sur l’existence globale pour le système de Navier-Stokes incompressible, SIAM J. Math. Anal., 23, 20-28 (1992) · Zbl 0762.35063
[11] Dai, M.; Liui, H., Long time behavior of solutions to the 3D Hall-magneto-hydrodynamics system with one diffusion, J. Differ. Equ., 266, 11, 7658-7677 (2019) · Zbl 1412.35248
[12] Dai, M., Local well-posedness of the Hall-MHD system in \(H^s( \mathbb{R}^n)\) with \(s > \frac{ n}{ 2} \), Math. Nachr., 293, 1, 67-78 (2020) · Zbl 1523.35122
[13] Danchin, R.; Tan, J., On the well-posedness of the Hall-magnetohydrodynamics system in critical spaces, Commun. Partial Differ. Equ. (2020)
[14] Danchin, R.; Tan, J., The global solvability of the Hall-magnetohydrodynamics system in critical Sobolev spaces (2019), arXiv e-prints
[15] Dumas, E.; Sueur, F., On the weak solutions to the Maxwell-Landau-Lifshitz equations and to the Hall-magneto-hydrodynamic equations, Commun. Math. Phys., 330, 1179-1225 (2014) · Zbl 1294.35094
[16] Gallagher, I.; Iftimie, D.; Planchon, F., Asymptotics and stability for global solutions to the Navier-Stokes equations, Ann. Inst. Fourier (Grenoble), 53, 1387-1424 (2003) · Zbl 1038.35054
[17] Huba, J. D.; Rudakov, L. I., Hall magnetohydrodynamics of neutral layers, Phys. Plasmas, 10, 3139-3150 (2003)
[18] Jeong, I.-J.; Oh, S.-J., On the Cauchy problem for the Hall and electron magnetohydrodynamic equations without resistivity I: illposedness near degenerate stationary solutions (2019), arXiv e-prints
[19] Li, J.; Yu, Y.; Zhu, W., A class large solution of the 3D Hall-magnetohydrodynamic equations, J. Differ. Equ., 268, 5811-5822 (2020) · Zbl 1434.35108
[20] Majda, A. J.; Bertozzi, A. L., Vorticity and Incompressible Flow (2001), Cambridge University Press
[21] Miao, C.; Yuan, B., On the well-posedness of the Cauchy problem for an MHD system in Besov spaces, Math. Methods Appl. Sci., 32, 53-76 (2009) · Zbl 1153.76066
[22] Paicu, M.; Zhang, P., Global solutions to the 3-D incompressible inhomogeneous Navier-Stokes system, J. Funct. Anal., 262, 3556-3584 (2012) · Zbl 1236.35112
[23] Peetre, J., New Thoughts on Besov Spaces, Duke University Mathematics Series, vol. 1 (1976), Mathematics Department, Duke University: Mathematics Department, Duke University Durham, NC · Zbl 0356.46038
[24] Somov, B. V., Magnetic reconnection in solar flares, Phys. Usp., 53, 954-958 (2010)
[25] Wan, R.; Zhou, Y., On global existence, energy decay and blow-up criteria for the Hall-MHD system, J. Differ. Equ., 259, 5982-6008 (2015) · Zbl 1328.35185
[26] Wan, R.; Zhou, Y., Global well-posedness for the 3D incompressible Hall-magnetohydrodynamic equations with Fujita-Kato type initial data, J. Math. Fluid Mech., 21, 1, Article 5 pp. (2019), 16 pp · Zbl 1414.35177
[27] Weng, S., On analyticity and temporal decay rates of solutions to the viscous resistive Hall-MHD system, J. Differ. Equ., 260, 6504-6524 (2016) · Zbl 1341.35133
[28] Weng, S., Space-time decay estimates for the incompressible viscous resistive MHD and Hall-MHD equations, J. Funct. Anal., 270, 2168-2187 (2016) · Zbl 1347.35207
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