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<p>This paper focuses on the Cauchy problem of the 3D MHD system with only horizontal dissipation. By using delicate energy estimates, we first establish blow-up criteria in terms of two components of the velocity and the magnetic field in Lorentz spaces. Moreover, we prove that this system is global well-posedness emanating from any initial data in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(H^s(\mathbb {R}^3)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mi>s</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(s\ge 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> provided the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-norm of the initial data and its vertical derivative are sufficiently small.</p>

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Global well-posedness of the 3D incompressible MHD system with only horizontal dissipation

  • 期刊论文

  • 发表日期：

    2026年7月10日

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  • Zhouyu Li,
  • Wenjing Song

摘要

This paper focuses on the Cauchy problem of the 3D MHD system with only horizontal dissipation. By using delicate energy estimates, we first establish blow-up criteria in terms of two components of the velocity and the magnetic field in Lorentz spaces. Moreover, we prove that this system is global well-posedness emanating from any initial data in \(H^s(\mathbb {R}^3)\) H s ( R 3 ) with \(s\ge 1\) s ≥ 1 provided the \(L^2\) L 2 -norm of the initial data and its vertical derivative are sufficiently small.