Moving One-Component Regularity Criteria for the 3D Incompressible MHD Equations

We establish a scaling-critical continuation criterion for the three-dimensional incompressible magnetohydrodynamic equations in ${\mathbb{R}}^3$ with arbitrary positive viscosity and magnetic diffusivity. Let $\beta (t)$ be a unit vector that is piecewise $H^1$ in time and has only finitely many jumps. For $3\le p<\mathrm{\infty }$, set ${\gamma }_p=2p\mathrm{/}(2p-3)$. We prove that an $H^1$ strong solution can be continued beyond $T_{\ast }$ if $${\int }_0^{T_{\ast }}(\mathrm{\parallel }u(t)\cdot \beta (t){\mathrm{\parallel }}_{{\dot{H}}^{3\mathrm{/}2}}^2+\mathrm{\parallel }b(t)\cdot \beta (t){\mathrm{\parallel }}_{{\dot{H}}^{3\mathrm{/}2}}^2+\mathrm{\parallel }\beta (t)\cdot \mathrm{curl}b(t){\mathrm{\parallel }}_{L^p}^{{\gamma }_p})dt<\mathrm{\infty }\mathrm{.}$$ Thus the observed component may vary in time, and the magnetic assumption is reduced to one moving magnetic component together with one Serrin-type current-density component. The proof is based on a moving-frame formulation of the vorticity-current system, an anisotropic product estimate adapted to the moving frame, and a horizontal Hodge decomposition that controls the current-Jacobian residual.