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.
Categories: math.AP