Abstract
In this paper, we consider the initial value problem of the incompressible Hall-MHD system and prove the global well-posedness in the scaling critical class \({\dot{B}}_{p,\infty }^{-1+\frac{3}{p}}(\mathbb {R}^3)\times ({\dot{B}}_{p,\infty }^{-1+\frac{3}{p}}(\mathbb {R}^3) \cap L^{\infty }(\mathbb {R}^3))\) for \(3< p < \infty \). Moreover, we also refine the smallness conditions and show that our global well-posedness holds for initial data whose \({\dot{B}}_{p,\infty }^{-1+\frac{3}{p}}(\mathbb {R}^3)\)-norm is large, provided that some weaker norm is sufficiently small.
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References
H. Bahouri, J.-Y. Chemin, and R. Danchin, Fourier analysis and nonlinear partial differential equations, Springer, Heidelberg, 2011.
H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Springer, New York, 2011.
D. Chae, P. Degond, and J.-G. Liu, Well-posedness for Hall-magnetohydrodynamics, Ann. Inst. H. Poincaré C Anal. Non Linéaire 31 (2014), 555–565.
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.
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.
R. Danchin and J. Tan, The Global Solvability of the Hall-MHD System in Critical Sobolev Spaces, arXiv:1912.09194.
M. Fujii and R. Nakasato, Dispersive phenomena for the incompressible Hall-MHD system around constant equilibrium states, preprint
H. Houamed, Well-posedness and long time behavior for the electron inertial Hall-MHD system in Besov and Kato-Herz spaces, J. Math. Anal. Appl. 501 (2021), no. 2, Paper No. 125208, 23, https://doi.org/10.1016/j.jmaa.2021.125208. MR4241621
T. Iwabuchi and M. Nakamura, Small solutions for nonlinear heat equations, the Navier–Stokes equation, and the Keller–Segel system in Besov and Triebel–Lizorkin spaces, Adv. Differential Equations 18 (2013), 687–736.
S. Kawashima, R. Nakasato, and T. Ogawa, Global well-posedness and time-decay of solutions for the compressible Hall-magnetohydrodynamic system in the critical Besov framework, J. Differential Equations 328 (2022), 1–64.
H. Kozono, T. Ogawa, and Y. Taniuchi, Navier–Stokes equations in the Besov space near\(L^\infty \)and BMO, Kyushu J. Math. 57 (2003), 303–324.
L. Liu and J. Tan, Global Well-posedness for the Hall-magnetohydrodynamics system in larger critical Besov spaces, HAL Id: hal-02430536.
R. Nakasato, Global well-posedness for the incompressible Hall-magnetohydrodynamic system in critical Fourier–Besov spaces, J. Evol. Equ. 22 (2022), Paper No. 20, 35.
J. Tan, New energy functionals for the incompressible Hall-MHD system. 2022. hal-03855638
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.
R. Wan and Y. Zhou, Global well-posedness for the 3D incompressible Hall-magnetohydrodynamic equations with Fujita–Kato type initial data, J. Math. Fluid Mech. 21 (2019), Paper No. 5, 16.
Acknowledgements
This work was supported by Grant-in-Aid for JSPS Research Fellow, Grant Number JP20J20941. The author would like to express his sincere gratitude to Professor Jun-ichi Segata, Faculty of Mathematics, Kyushu University, for many fruitful advices and continuous encouragement.
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Fujii, M. Global well-posedness of the incompressible Hall-MHD system in critical spaces. J. Evol. Equ. 24, 3 (2024). https://doi.org/10.1007/s00028-023-00933-8
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DOI: https://doi.org/10.1007/s00028-023-00933-8