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 $L^4_{t,x}$ 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 $8π$.
math.AP↗