2022/10/11 by Juncheng Wei, Qidi Zhang, Wei, Juncheng +3
Economics, Econometrics and Finance · Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics and Turbulent Flows #Mathematical Physics (math-ph) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2210.05800
openalex publication_date 2022/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We construct finite time blow-up solutions to the Landau-Lifshitz-Gilbert equation (LLG) from \mathbb R2 into S2 \begincases ut= a(Δu+|∇ u|2u) -b u\wedge Δu amp; in \mathbb R2×(0,T), u(⋅,0) = u0∈ S2 amp; in \mathbb R2, \endcases where a2+b2=1,~a > 0,~ b∈ \mathbb R. Given any prescribed N points in ℝ2 and small T>0, we prove that there exists regular initial data such that the solution blows up precisely at these points at finite time t=T, taking around each point the profile of sharply scaled degree 1 harmonic map with the type II blow-up speed ‖ ∇ u‖L^∞ ∼ (|ln(T-t)|2)/( T-t ) as t→ T. The proof is based on the \em parabolic inner-outer gluing method, developed in \cite17HMF for Harmonic Map Flow (HMF). However, a direct consequence of the presence of dispersion is the \em lack of maximum principle for suitable quantities, which makes the analysis more delicate even at the linearized level. To overcome this difficulty, we make use of two key technical ingredients: first, for the inner problem we employ the tool of \em distorted Fourier transform, as developed by Krieger, Miao, Schlag and Tataru \citeKrieger09Duke,KMS20WM. Second, the linear theory for the outer problem is achieved by means of the sub-Gaussian estimate for the fundamental solution of parabolic system in non-divergence form with coefficients of Dini mean oscillation in space (DMOx), which was proved by Dong, Kim and Lee \citedong22-non-divergence recently.