Beale–Kato–Majda-type continuation criterion for free boundary MHD

Let $d$ be the spatial dimension, let $\frac{d}{2}+1<\sigma<\infty$, and let $(v,B,\Gamma)\in C([0,T);\mathbf{H}^{\sigma})$ be a solution to the free boundary MHD equations. Write $a(t)$ for the Taylor coefficient, let $\mathbf{C}^{\alpha}$ denote the natural Hölder space analogue of $\mathbf{H}^{\alpha}$, and let the solution remain in the collar. Continuation criterion. There exists $1\leq p\leq\infty$ such that $(v,B,\Gamma)$ can be continued past time $T>0$ as long as there is a $c>0$ with

$$a(t)\geq c>0,\qquad 0\leq t<T,$$

and

$$\|(v,B,\Gamma)\|_{L^{\infty}([0,T);\mathbf{C}^{\frac{1}{2}+\epsilon})}+\|(v,B,\Gamma)\|_{L^{p}([0,T);\mathbf{C}^{1})}<\infty.$$

This is a conjectural Beale–Kato–Majda-type criterion seeking continuation under only Hölder-based bounds on the natural variables, together with a uniform positive lower bound for the Taylor coefficient. Establishing it would require further optimization of the paper’s energy estimates and more delicate balanced elliptic estimates; it remains open.