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<p>This paper establishes a priori estimates for the free boundary problem of two-dimensional ideal incompressible magnetohydrodynamics (MHD) involving a plasma-vacuum interface. The system consists of a plasma region governed by the ideal MHD equations and a vacuum region described by pre-Maxwell dynamics, separated by a freely evolving interface where total pressure continuity and magnetic field tangency conditions hold. By adopting a geometric Lagrangian framework, we reformulate the free boundary problem into a fixed domain using trajectory maps and fictitious velocity extensions. Our main contribution lies in deriving higher-order energy norms that combine boundary geometry and interior dynamics, enabling control over the solution’s regularity. For the vacuum magnetic field, we prove that its covariant derivatives in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> norms are bounded solely by initial data and curvature parameters, leveraging geometric trace inequalities under bounded curvature constraints. Additionally, we establish evolution equations for the electric field in vacuum and demonstrate energy exchanges between plasma and vacuum regions through pressure balance. The results provide a crucial step towards proving well-posedness for this challenging interface problem.</p>

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A Priori Estimates of Plasma-Vacuum Interface Problem for 2D Incompressible Ideal MHD

  • 期刊论文

  • 发表日期：

    2026年7月2日

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  • Chengchun Hao

摘要

This paper establishes a priori estimates for the free boundary problem of two-dimensional ideal incompressible magnetohydrodynamics (MHD) involving a plasma-vacuum interface. The system consists of a plasma region governed by the ideal MHD equations and a vacuum region described by pre-Maxwell dynamics, separated by a freely evolving interface where total pressure continuity and magnetic field tangency conditions hold. By adopting a geometric Lagrangian framework, we reformulate the free boundary problem into a fixed domain using trajectory maps and fictitious velocity extensions. Our main contribution lies in deriving higher-order energy norms that combine boundary geometry and interior dynamics, enabling control over the solution’s regularity. For the vacuum magnetic field, we prove that its covariant derivatives in \(L^2\) L 2 norms are bounded solely by initial data and curvature parameters, leveraging geometric trace inequalities under bounded curvature constraints. Additionally, we establish evolution equations for the electric field in vacuum and demonstrate energy exchanges between plasma and vacuum regions through pressure balance. The results provide a crucial step towards proving well-posedness for this challenging interface problem.