Exact boundary controllability of the 3D incompressible ideal MHD system

TL;DR

This paper establishes the boundary controllability of the 3D ideal MHD system in bounded domains, proving that arbitrary divergence-free states can be reached via boundary controls, and introduces a new local well-posedness proof without Elsasser variables.

Contribution

It proves the first boundary controllability results for the 3D ideal MHD system and provides a novel local well-posedness proof applicable to any bounded domain.

Findings

Boundary controllability of 3D ideal MHD system established.

First local well-posedness proof without Elsasser variables.

New simple proof of 2D controllability.

Abstract

We consider the three-dimensional ideal MHD system on a domain $\Omega' \subset \mathbb{R}^3$ with a part $\Gamma$ of the boundary~$\partial \Omega$, where we prescribe both $u\cdot n$ and $b\cdot n$, while $u\cdot n = b\cdot n =0$ on $\partial \Omega' \setminus \Gamma$. We prove the boundary controllability of the system, namely that we can prescribe the boundary data such that the unique solution of the system with initial state $(u_0,b_0)$ achieves another state $(u_1,b_1)$ in finite time, where $u_0,b_0,u_1,b_1$ are arbitrary divergence-free vector fields satisfying impermeability boundary condition which are extendable to vector fields with the same properties on any bounded domain obtained by extension of $\Omega'$ via $\Gamma$. As a byproduct, we give the first local well-posedness proof of incompressible, ideal MHD system, which does not use Elsasser variables and is thus applicable to any bounded domain with sufficient Sobolev regularity. We also provide a new and simple proof of the $2$D controllability.