On the Grad-Rubin boundary value problem for the two-dimensional magneto-hydrostatic equations

TL;DR

This paper investigates the solvability of a boundary value problem for 2D magneto-hydrostatic equations using fixed point methods, extending techniques to related Euler equations.

Contribution

It introduces a novel fixed point approach combining current transport and singular integral estimates for magneto-hydrostatic and Euler boundary problems.

Findings

Established solvability of the boundary value problem.

Extended method to steady incompressible Euler equations.

Provided new analytical tools for non-convolution singular integrals.

Abstract

In this work, we study the solvability of a boundary value problem for the magneto-hydrostatic equations originally proposed by Grad and Rubin in the late 50's. The proof relies on a fixed point argument which combines the so-called current transport method together with H\"older estimates for a class of non-convolution singular integral operators. The same method allows to solve an analogous boundary value problem for the steady incompressible Euler equations.