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

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 ${\mathbf{u}}_0\in B_{2,1}^{n/2-1}$ and ${\mathbf{B}}_0\in B_{2,1}^{n/2}$. The a priori estimates include the term ${\int }_0^t\Vert \mathbf{u}(s){\Vert }_{H^{n/2}}^2\mathrm{d}s$ 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 Calderón (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.