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<p>In this paper, we investigate several regularity criteria to the 3D incompressible magnetohydrodynamic equations. These criteria are based on certain assumptions made in partial elements of the velocity gradient tensor and pressure, respectively. By making use of the Littlewood-Paley decomposition, we show that the solution (<i>u, b</i>) can be smoothly extended after time <i>T</i> if any two groups of functions (<i>∂</i><sub>1</sub><i>u</i><sub>1</sub>, <i>∂</i><sub>1</sub><i>b</i><sub>1</sub>), (<i>∂</i><sub>2</sub><i>u</i><sub>2</sub>, <i>∂</i><sub>2</sub><i>b</i><sub>2</sub>) and (<i>∂</i><sub>3</sub><i>u</i><sub>3</sub>, <i>∂</i><sub>3</sub><i>b</i><sub>3</sub> belong to the space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^{1}(0,T;\dot{B}_{\infty,\infty}^0)\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mi>L</mi> <mrow> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo>;</mo> <msubsup> <mrow> <mover> <mi>B</mi> <mo>˙</mo> </mover> </mrow> <mrow> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> </mrow> <mn>0</mn> </msubsup> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation>.</p>

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Some new regularity criteria of the incompressible 3D MHD equations

  • 期刊论文

  • 发表日期：

    2026年5月9日

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  • Honglin Zhang,
  • Li Li,
  • Qingning Zhang

摘要

In this paper, we investigate several regularity criteria to the 3D incompressible magnetohydrodynamic equations. These criteria are based on certain assumptions made in partial elements of the velocity gradient tensor and pressure, respectively. By making use of the Littlewood-Paley decomposition, we show that the solution (u, b) can be smoothly extended after time T if any two groups of functions (∂1u1, ∂1b1), (∂2u2, ∂2b2) and (∂3u3, ∂3b3 belong to the space \(L^{1}(0,T;\dot{B}_{\infty,\infty}^0)\) L 1 ( 0 , T ; B ˙ ∞ , ∞ 0 ) .