Regularity criteria for the three-dimensional MHD equations

Luo, Lan; Zhao, Yongye; Yang, Qing

doi:10.1007/s10255-011-0108-4

Cited by 5 publications

(5 citation statements)

References 12 publications

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“…Therefore, a large number of articles appear to discuss this subject by addressing the sufficient conditions with which to guarantee the global regularity of the weak solution. Different criteria for regularity in terms of the velocity field, the magnetic field, the pressure and their derivatives have been proposed (see [2,7,8,[13][14][15][16][17][19][20][21][22][23][24] and references therein).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…It follows from Gronwall's inequality that with α > 7 2 and β 1. Hence, using the regularity criterion (72) in [17], the proof is complete for α > 7 2 . Next, consider the case of α The right-hand side of the inequality (90) is bounded, provided that In fact, by choosing s > α one can easily see that 2α(s−1) s(α−1 ) 2.…”

Section: Regularity Criterion For Pressurementioning

confidence: 88%

“…Note that to match the condition 2 r 3 in the inequality (17), we have to restrict the index α 2. Furthermore, it follows from Young's inequality that…”

Section: Regularity Criterion On the Derivatives Of The Velocity In O...mentioning

confidence: 99%

“…Proof of theorem 1.4. To obtain the regularity criterion for pressure, we will use the recent results on the regularity criteria of three-dimensional MHD in [17]. In that paper, Luo…”

Section: Regularity Criterion For Pressurementioning

confidence: 99%

See 3 more Smart Citations

Regularity criteria for incompressible magnetohydrodynamics equations in three dimensions

Lin¹,

Li²

2012

Nonlinearity

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In this paper, we give some new global regularity criteria for three-dimensional incompressible magnetohydrodynamics (MHD) equations. More precisely, we provide some sufficient conditions in terms of the derivatives of the velocity or pressure, for the global regularity of strong solutions to 3D incompressible MHD equations in the whole space, as well as for periodic boundary conditions. Moreover, the regularity criterion involving three of the nine components of the velocity gradient tensor is also obtained. The main results generalize the recent work by Cao and Wu (2010 Two regularity criteria for the 3D MHD equations J. Diff. Eqns 248 2263-74) and the analysis in part is based on the works by Cao C and Titi E (2008 Regularity criteria for the three-dimensional Navier-Stokes equations Indiana Univ. Math. J. 57 2643-61; 2011 Gobal regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor Arch. Rational Mech. Anal. 202 919-32) for 3D incompressible Navier-Stokes equations.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Regularity Criterion For Pressurementioning

confidence: 88%

“…Note that to match the condition 2 r 3 in the inequality (17), we have to restrict the index α 2. Furthermore, it follows from Young's inequality that…”

Section: Regularity Criterion On the Derivatives Of The Velocity In O...mentioning

confidence: 99%

“…Proof of theorem 1.4. To obtain the regularity criterion for pressure, we will use the recent results on the regularity criteria of three-dimensional MHD in [17]. In that paper, Luo…”

Section: Regularity Criterion For Pressurementioning

confidence: 99%

See 2 more Smart Citations

Regularity criteria for incompressible magnetohydrodynamics equations in three dimensions

Lin¹,

Li²

2012

Nonlinearity

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“…For example, He-Xin[9] proved that if is a weak solution of (1.1), and satisfies (1.8), then is smooth in , and of course it satisfies the related energy equality: for all . For more results in this field, please see [2, 8, 16, 17, 21]. However, there is a little literature on energy conservation criteria for the standard MHD equations, the only reference, to our knowledge, is [12] where the author proved if the pair is a weak solution of the MHD equations and fulfills energy conservation criteria (1.7), then the energy equality (1.9) holds.…”

Section: Introductionmentioning

confidence: 99%

On the energy equality for very weak solutions to 3D MHD equations

Lai¹,

Yang²

2021

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In this paper, we consider the energy equality of the 3D Cauchy problem for the magneto-hydrodynamics (MHD) equations. We show that if a very weak solution of MHD equations belongs to $L^{4}(0,\,T;L^{4}(\mathbb {R}^{3}))$ , then it is actually in the Leray–Hopf class and therefore must satisfy the energy equality in the time interval $[0,\,T]$ .

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The Regularity of Very Weak Solutions to Magneto-Hydrodynamics Equations

Lai

Tang

2023

J. Math. Fluid Mech.

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Regularity criteria for the three-dimensional MHD equations

Cited by 5 publications

References 12 publications

Regularity criteria for incompressible magnetohydrodynamics equations in three dimensions

Regularity criteria for incompressible magnetohydrodynamics equations in three dimensions

On the energy equality for very weak solutions to 3D MHD equations

The Regularity of Very Weak Solutions to Magneto-Hydrodynamics Equations

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