Continuation and threshold global strong solutions for the two-dimensional Landau--Lifshitz--Gilbert equation

This paper studies finite-terminal continuation and subthreshold global existence for the two-dimensional Landau–Lifshitz–Gilbert equation in the $H_Q^2$ strong-solution class. The proof shows that if a maximal strong solution terminates at a finite time and the full-frequency $L_{t,x}^4$ norm of $\mathrm{\nabla }u$ is finite on a terminal time window, then the strong solution can be continued. Furthermore, the solution can still be continued as long as the ultraviolet tail above one fixed dyadic cutoff belongs to $L_{t,x}^4$. As applications, we obtain global strong solutions when $E(u_0)<4\pi$; in the degree-zero sector, this energy threshold can be raised to $8\pi$.