A Priori Estimates for Free Boundary Problem of Incompressible Inviscid Magnetohydrodynamic Flows

Abstract

Abstract. In the present paper, we prove the a priori estimates of Sobolev norms for a free boundary problem of the incompressible inviscid MHD equations in all physical spatial dimensions n = 2 and 3 by adopting a geometrical point of view used in [4], and estimating quantities such as the second fundamental form and the velocity of the free surface. We identify the well-posedness condition that the outer normal derivative of the total pressure including the fluid and magnetic pressures is negative on the free boundary, which is similar to the physical condition (Taylor sign condition) for the incompressible Euler equations of fluids.

Nonlinear Stability of the Current‐Vortex Sheet to the Incompressible MHD Equations

“…We believe that our method can be applied to solve the plasma-vacuum interface problem in ideal incompressible MHD and the other related free boundary problems (see [14], for example). We note that the well-posedness of the linearized plasma-vacuum interface problem has been proved by Morando, Trakhinin, and Trebeschi [20].…”

A regularity result for the incompressible magnetohydrodynamics equations with free surface boundary

“…One may drop the requirement for η H s+0.5 when s > 3.5 using Alinhac's good unknowns thanks to the fact that ∂a ∈ L ∞ . We refer [14,16] for details.…”

Well-posedness of the Free Boundary Problem for Non-relativistic and Relativistic Compressible Euler Equations with a Vacuum Boundary Condition

“…This problem is a nonlinear hyperbolic problem with a free characteristic boundary due to condition (1.5a). Assumption (1.6) is also the natural physical condition for the incompressible MHD; see Hao-Luo [14] for a priori estimates through a geometrical point of view and Gu-Wang [12] for local well-posedness. We also refer to Hao-Luo [15] for a recent ill-posedness result of the 2D incompressible MHD when condition (1.6) is violated.…”