Local interior regularity for the 3D MHD equations in nonendpoint borderline Lorentz space

Jae-Myoung Kim; Jae-Myoung Kim

doi:10.3934/math.2021148

AIMS Mathematics

2021, Volume 6, Issue 3: 2440-2453. doi: 10.3934/math.2021148

Previous Article Next Article

Research article

Local interior regularity for the 3D MHD equations in nonendpoint borderline Lorentz space

Jae-Myoung Kim ^,

Department of Mathematics Education, Andong National University, Andong 36729, Republic of Korea

Received: 14 November 2020 Accepted: 11 December 2020 Published: 21 December 2020
MSC : 35B65, 76W05

We prove local regularity condition for a suitable weak solution to 3D MHD equations. Precisely, if a solution satisfies $u, b \in L^{\infty}(-(\frac{4}{3})^2, 0;L^{3, q}(B_{\frac{3}{4}}))$, $q\in (3, \infty)$ in Lorentz space, then $(u, b)$ is Hölder continuous in the closure of the set $Q_{\frac{1}{2}}$.
- local regularity condition,
- suitable weak solution,
- 3D MHD equations
Citation: Jae-Myoung Kim. Local interior regularity for the 3D MHD equations in nonendpoint borderline Lorentz space[J]. AIMS Mathematics, 2021, 6(3): 2440-2453. doi: 10.3934/math.2021148

Related Papers:

Abstract

We prove local regularity condition for a suitable weak solution to 3D MHD equations. Precisely, if a solution satisfies $u, b \in L^{\infty}(-(\frac{4}{3})^2, 0;L^{3, q}(B_{\frac{3}{4}}))$, $q\in (3, \infty)$ in Lorentz space, then $(u, b)$ is Hölder continuous in the closure of the set $Q_{\frac{1}{2}}$.

References

[1]	R. P. Agarwal, S. Gala, M. A. Ragusa, A regularity criterion in weak spaces to Boussinesq Equations, Mathematics, 8 (2020), 920. doi: 10.3390/math8060920
[2]	T. Barker, Local boundary regularity for the Navier-Stokes equations in nonendpoint borderline Lorentz spaces, J. Math. Sci.(N.Y.), 224 (2017), 391-413. doi: 10.1007/s10958-017-3424-2
[3]	S. Benbernou, S. Gala, M. A. Ragusa, On the regularity criteria for the 3D magnetohydrodynamic equations via two components in terms of $BMO$ space, Math. Methods Appl. Sci., 37 (2014), 2320-2325. doi: 10.1002/mma.2981
[4]	L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831. doi: 10.1002/cpa.3160350604
[5]	P. A. Davidson, An introduction to magnetohydrodynamics, Cambridge University Press, Cambridge, 2001.
[6]	L. Escauriaza, G. Seregin, V. Šverák, $L^{3, \infty}$-solutions of Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk, 58 (2003), 211-250.
[7]	N. S. Khan, Mixed convection in MHD second grade nanofluid flow through a porous medium containing nanoparticles and gyrotactic microorganisms with chemical reaction, Filomat, 33 (2019), 4627-4653. doi: 10.2298/FIL1914627K
[8]	C. He, Z. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Funct. Anal., 227 (2005), 113-152. doi: 10.1016/j.jfa.2005.06.009
[9]	O. A. Ladyženskaja, V. A. Solonnikov, Solution of some non-stationary problems of magnetohydrodynamics for a viscous incompressible fluid, Trudy Mat. Inst. Steklov., Acad. Sci. USSR, Moscow-Leningrad, 59 (1960), 115-173.
[10]	O. A. Ladyženskaja, G. A. Seregin, On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations, J. Math. Fluid Mech., 1 (1999), 356-387. doi: 10.1007/s000210050015
[11]	F. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math., 51 (1998), 241-257. doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A
[12]	Y. Luo, T. P. Tsai, Regularity criteria in weak $L^3$ for 3D incompressible Navier-Stokes equations, Funccialaj Ekvacioj, Comm. Pure Appl. Math., 58 (2015), 387-404.
[13]	A. Mahalov, A. Nicolaenko, A. Shilkin, $L^{3, \infty}$-solutions to the MHD equations, J. Math. Sci. (N. Y.), 143 (2007), 2911-2923. doi: 10.1007/s10958-007-0175-5
[14]	N. C. Phuc, The Navier-Stokes equations in nonendpoint borderline Lorentz spaces, J. Math. Fluid Mech., 17 (2015), 741-760. doi: 10.1007/s00021-015-0229-2
[15]	M. Sermange, R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506
[16]	G. A. Seregin, Local regularity of suitable weak solutions to the Navier-Stokes equations near the boundary, J. Math. Fluid Mech., 4 (2002), 1-29. doi: 10.1007/s00021-002-8533-z
[17]	G. A. Seregin, Some estimates near the boundary for solutions to the non-stationary linearized Navier-Stokes equations, J. Math. Sci. (N. Y.), 115 (2003), 2820-2831. doi: 10.1023/A:1023330105200
[18]	G. A. Seregin, On smoothness of $L_{3, \infty}$-solutions to the Navier-Stokes equations up to boundary, Math. Ann., 332 (2005), 219-238. doi: 10.1007/s00208-004-0625-z
[19]	G. A. Seregin, A note on local boundary regularity for the Stokes system, J. Math. Sci. (N.Y.), 166 (2010), 86-90. doi: 10.1007/s10958-010-9847-7
[20]	S. Takahashi, On interior regularity criteria for weak solutions of the Navier-Stokes equations, Manuscripta Math., 69 (1990), 237-254. doi: 10.1007/BF02567922
[21]	H. Triebel, Theory of Function Spaces, Birkhäuser Verlag. Basel-Boston, (1983).
[22]	V. Vyalov, T. Shilkin, Partial regularity of solutions to the magnetohydrodynamic equations, J. Math. Sci. (N. Y.), 150 (2008), 1771-1786. doi: 10.1007/s10958-008-0095-z
[23]	W. Wang, Z. Zhang, Limiting case for the regularity criterion to the 3-D magneto-hydrodynamics equations, J. Differential Equations, 252 (2012), 5751-5762. doi: 10.1016/j.jde.2012.01.043

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)