BEALE-KATO-MAJDA TYPE BLOW-UP CRITERIA OF SMOOTH SOLUTIONS TO THE 3D INCOMPRESSIBLE (HALL)-MHD EQUATIONS IN BOUNDED DOMAINS
Keywords:
Hall-Magnetohydrodynamics equations, Brezis-Gallouet-Wainger inequality, regularity criteriaDOI:
https://doi.org/10.17654/0972087124016Abstract
We study Beale-Kato-Majda type blow-up criteria of smooth solutions to the 3D incompressible (Hall)-MHD equations on a general bounded domain in $\R^3$ based on a fine version of the Brezis-Gallouet-Wainger inequality.
Received: June 26, 2024
Revised: July 23, 2024
Accepted: August 2, 2024
References
M. Acheritogaray, P. Degond, A. Frouvelle and J.-G. Liu, Kinetic formulation and global existence for the Hall-magnetohydrodynamics system, Kinet. Relat. Models. 4 (2012), 901-918.
D. Chae, P. Degond and J.-G. Liu, Well-posedness for Hall-magnetohydrodynamics, Ann. Inst. H. Poincare C Anal. Non Lineaire 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.
D. Chae and M. Schonbek, On the temporal decay for the Hall-magnetohydrodynamic equation, J. Differential Equations 255 (2013), 3971-3982.
D. Chae and J. Wolf, On partial regularity for the 3D nonstationary Hall- magnetohydrodynamics equations on the plane, SIAM J. Math. Anal. 48 (2016), 443-469.
J. Fan, Y. Fukumoto, G. Nakamura and Y. Zhou, Regularity criteria for the incompressible Hall-MHD system, ZAMM Z. Angew. Math. Mech. 95 (2015), 1156-1160.
J. Fan, F. Li and G. Nakamura, Regularity criteria for the incompressible Hall-magnetohydrodynamic equations, Nonlinear Anal. 109 (2014), 173-179.
F. Wu, Navier-Stokes regularity criteria in Vishik spaces, Applied Mathematics and Optimization 84 (2021), 39-53.
R. Farwig and R. Kanamaru, Optimality of Serrin type extension criteria to the Navier-Stokes equations, Adv. Nonlinear Anal. 10 (2021), 1071-1085.
Y. Giga and H. Sohr, Abstract estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal. 102 (1991), 72-94.
K. Kang and J.-M. Kim, Regularity criteria of the magnetohydrodynamic equations in bounded domains or a half space, J. Differential Equations 253 (2012), 764-794.
R. Kanamaru, Optimality of logarithmic interpolation inequalities and extension criteria to the Navier-Stokes and Euler equations in Vishik spaces, J. Evol. Equ. 20(4) (2020), 1381-1397.
K. Nakao and Y. Taniuchi, Brezis-Gallouet-Wainger type inequalities and blow-up criteria for Navier-Stokes equations in unbounded domains, Comm. Math. Phys. 359 (2018), 951-973.
T. Ogawa and Y. Taniuchi, On a blow-up criterion for the 3-D Euler equations in a bounded domain, Nonlinear Evolution Equations and Applications (Japanese) (Kyoto, 2000), Surikaisekikenkyusho Kokyuroku No. 1197, 2001, pp. 60-72.
K. Ryo, Brezis-Gallouet-Wainger type inequalities and a priori estimates of strong solutions to Navier-Stokes equations, J. Funct. Anal. 278 (2020), 108277.
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.
L. Stupelis, An initial boundary-value problem for a system of equations of magnetohydrodynamics, Lithuanian Math. J. 40 (2000), 176-196.
R. Shimada, On the maximal regularity for Stokes equations with Robin boundary condition in a bounded domain, Math. Methods Appl. Sci. 30 (2007), 257-289.
A. Takahiro, On the existence of solutions of the magnetohydrodynamic equations in a bounded domain, Nonlinear Anal. 54(6) (2003), 1165-1174.
M. Vishik, Incompressible flows of an ideal fluid with unbounded vorticity, Comm. Math. Phys. 213 (2000), 697-731.
S. Weng, On analyticity and temporal decay rates of solutions to the viscous resistive Hall-MHD system, J. Differential Equations 260 (2016), 6504-6524.
Z. Ye, Regularity criterion for the 3D Hall-magnetohydrodynamic equations involving the vorticity, Nonlinear Anal. 144 (2016), 182-193.
W. von Wahl, Estimating by div u and curl, Math. Methods Appl. Sci. 15 (1992), 123-143.
Z. Wen, An improved regularity criterion for the 3D Hall-MHD equations via the vorticity, Comput. Math. Appl. 75 (2018), 821-836.
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