A Regularity Result for the Incompressible Magnetohydrodynamics Equations with Free Surface Boundary

Abstract

We consider the three-dimensional incompressible magnetohydrodynamics (MHD) equations in a bounded domain with small volume and free moving surface boundary. We establish a priori estimate for solutions with minimal regularity assumptions on the initial data in Lagrangian coordinates. In particular, due to the lack of the Cauchy invariance for MHD equations, the smallness assumption on the fluid domain is required to compensate a loss of control of the flow map. Moreover, we show that the magnetic field has certain regularizing effect which allows us to control the vorticity of the fluid and that of the magnetic field. To the best of our knowledge this is the first result that focuses on the low regularity solution for incompressible free-boundary MHD equations.

Final version, to appear in Nonlinearity

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Cited by in corpus (9)

• Well-posedness for the Linearized Free Boundary Problem of Incompressible Ideal Magnetohydrodynamics Equations
• Local Well-posedness of the Free-Boundary Incompressible Magnetohydrodynamics with Surface Tension
• Zero Surface Tension Limit of the Free-Boundary Problem in Incompressible Magnetohydrodynamics
• A priori Estimates for the Incompressible Free-Boundary Magnetohydrodynamics Equations with Surface Tension
• Local Well-posedness of the Free-Boundary Problem in Compressible Resistive Magnetohydrodynamics
• Anisotropic Regularity of the Free-Boundary Problem in Compressible Ideal Magnetohydrodynamics
• Splash singularity for the free boundary incompressible viscous MHD
• A priori Estimates for the Free-Boundary problem of Compressible Resistive MHD Equations and Incompressible Limit
• A priori estimates and a blow-up criterion for the incompressible ideal MHD equations with surface tension and a closed free surface