AI 中文总结
本文证明二维Landau--Lifshitz--Gilbert方程在$H_Q^2$类中，基于全频或紫外尾部$L^4_{t,x}$范数条件可实现有限时间延拓，并由此得到能量阈值$4\pi$（零度扇区$8\pi$）下的全局强解。
AI 中文摘要
本文研究二维Landau--Lifshitz--Gilbert方程在$H_Q^2$强解类中的有限终端延拓和亚阈值全局存在性。证明表明，若一个极大强解在有限时间终止，且$\nabla u$的全频$L^4_{t,x}$范数在终端时间窗口内有限，则该强解可以延拓。此外，只要高于一个固定二进截断的紫外尾部属于$L^4_{t,x}$，解仍可延拓。作为应用，当$E(u_0)<4\pi$时，我们获得全局强解；在零度扇区中，此能量阈值可提升至$8\pi$。
英文摘要
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 $L^4_{t,x}$ norm of $\nabla u$ 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 $L^4_{t,x}$. As applications, we obtain global strong solutions when $E(u_0)<4π$; in the degree-zero sector, this energy threshold can be raised to $8π$.
