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<p>This paper studies the initial boundary value problems for the 2D MHD equations with vertical dissipation and horizontal magneto damping in a strip domain. The proof is nontrivial and relies on a sophisticated analytical framework involving anisotropic norms, time-derivative estimates, and time-weighted energy functionals. To address the challenges posed by weak dissipation and boundary effects, we develop a set of anisotropic inequalities and establish key auxiliary estimates. For sufficiently small initial data, a careful analysis of the nonlinear terms then leads to the global well-posedness of the system in the Sobolev setting <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(H^3\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation>. Furthermore, the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(H^{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-norm of the velocity field and the magnetic field are shown to decay exponentially in time.</p>

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Initial-Boundary Value Problems for the 2D MHD Equations with Partial Dissipation

  • 期刊论文

  • 发表日期：

    2026年7月21日

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  • Quansen Jiu,
  • Xiaoxiao Suo

摘要

This paper studies the initial boundary value problems for the 2D MHD equations with vertical dissipation and horizontal magneto damping in a strip domain. The proof is nontrivial and relies on a sophisticated analytical framework involving anisotropic norms, time-derivative estimates, and time-weighted energy functionals. To address the challenges posed by weak dissipation and boundary effects, we develop a set of anisotropic inequalities and establish key auxiliary estimates. For sufficiently small initial data, a careful analysis of the nonlinear terms then leads to the global well-posedness of the system in the Sobolev setting \(H^3\) H 3 . Furthermore, the \(H^{2}\) H 2 -norm of the velocity field and the magnetic field are shown to decay exponentially in time.