Far East Journal of Mathematical Sciences (FJMS)

The Far East Journal of Mathematical Sciences (FJMS) publishes original research papers and survey articles in pure and applied mathematics, statistics, mathematical physics, and other related fields. It welcomes application-oriented work as well.

Submit Article

BEALE-KATO-MAJDA TYPE BLOW-UP CRITERIA OF SMOOTH SOLUTIONS TO THE 3D INCOMPRESSIBLE (HALL)-MHD EQUATIONS IN BOUNDED DOMAINS

Authors

  • Jaewoo Kim
  • Jae-Myoung Kim

Keywords:

Hall-Magnetohydrodynamics equations, Brezis-Gallouet-Wainger inequality, regularity criteria

DOI:

https://doi.org/10.17654/0972087124016

Abstract

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.

Published

2024-08-14

Issue

Section

Articles

How to Cite

BEALE-KATO-MAJDA TYPE BLOW-UP CRITERIA OF SMOOTH SOLUTIONS TO THE 3D INCOMPRESSIBLE (HALL)-MHD EQUATIONS IN BOUNDED DOMAINS. (2024). Far East Journal of Mathematical Sciences (FJMS), 141(4), 267-281. https://doi.org/10.17654/0972087124016

Similar Articles

1-10 of 22

You may also start an advanced similarity search for this article.