Local existence for the non-resistive MHD equations in Besov spaces

TL;DR

This paper establishes local existence of solutions to the non-resistive MHD equations in Besov spaces for 2D and 3D, with a focus on the necessary a priori estimates and the challenges in proving uniqueness.

Contribution

It provides the first proof of local existence in Besov spaces for the non-resistive MHD equations, using Calderón's splitting method for 3D estimates.

Findings

Existence of solutions in Besov spaces for 2D and 3D MHD equations.

In 3D, auxiliary estimates in H^{1/2} are achieved using Calderón's splitting method.

Uniqueness of solutions is proven in 3D, but remains open in 2D.

Abstract

In this paper we prove the existence of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations on the whole of $\mathbb{R}^{n}$, $n=2,3$, for divergence-free initial data in certain Besov spaces, namely $\boldsymbol{u}_{0} \in B^{n/2-1}_{2,1}$ and $\boldsymbol{B}_{0} \in B^{n/2}_{2,1}$. The a priori estimates include the term $\int_{0}^{t} \| \boldsymbol{u}(s) \|_{H^{n/2}}^{2} \, \mathrm{d} s$ on the right-hand side, which thus requires an auxiliary bound in $H^{n/2-1}$. In 2D, this is simply achieved using the standard energy inequality; but in 3D an auxiliary estimate in $H^{1/2}$ is required, which we prove using the splitting method of Calder\'on (Trans. Amer. Math. Soc. 318(1), 179--200, 1990). By contrast, we prove that such solutions are unique in 3D, but the proof of uniqueness in 2D is more difficult and remains open.