Finite-time singularity formations for the Landau-Lifshitz-Gilbert equation in dimension two

Abstract

We construct finite time blow-up solutions to the Landau-Lifshitz-Gilbert equation (LLG) from R 2 into S 2 ut = a(∆u + |∇u| 2 u) − bu ∧ ∆u in R 2 × (0, T ),whereGiven any prescribed N points in R 2 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 speedThe proof is based on the parabolic inner-outer gluing method, developed in [12] for Harmonic Map Flow (HMF). However, substantial difficulties arise due to the coupling between HMF and Schrodinger Map Flow (SMF) in LLG, and such coupling produces both dissipative (a > 0) and dispersive (b = 0) features. A direct consequence of the presence of dispersion is the lack of maximum principle for suitable quantities, which makes the analysis more delicate even at the linearized level. The dispersion cannot be treated perturbatively even in the dissipation-dominating case a/|b| >> 1, and one has to include this as part of the leading order. To overcome these difficulties, we make use of two key technical ingredients: first, for the inner problem we employ the tool of distorted Fourier transform, as developed by Krieger, Miao, Schlag and Tataru [34,35]. 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 [20] recently. J. WEI, Q. ZHANG, AND Y. ZHOU 9.3. Higher modes |k| ≥ 2 71 9.4. Mode 0 77 9.5. Mode 1 81 9.6. Mode −1 86 Appendix A. Some useful estimates 95 Appendix B. Convolution estimates in finite time 101 B.1. Preliminary 101 B.2. Convolution about v(t)|x − q| −b 1 {l1(t)≤|x−q|≤l2(t)} 110 B.3. Convolution about v(t)|x − q| −b 1 {|x−q|≥(T −t) 1 2 } 120 Appendix C. Derivation of the weighted topology for the outer problem 123 Appendix D. Estimates of G and H j 129 D.1. Estimates for terms involving Φ out , Φ [j] in 129 D.2. Estimate of ∇ x A 132 D.3. Estimate of G 136 D.4. Estimate of H [j] 161 Acknowledgements 166 References 166

An Adaptive Moving Mesh Method for Simulating Finite-Time Blowup Solutions of the Landau-Lifshitz-Gilbert Equation

“…In this section, our aim is to compare confirm our numerical results in the preceding section, with the analytical results in a recent study. The finite time singularity formulation for the Landau-Lifshitz-Gilbert equations is studied in [30]. The authors consider the following Landau-Lifshitz-Gilbert equation from 2 to S 2 : The authors in [30] prove that for any given N points in R 2 and small T > 0, there exists regular initial data such that the solution of (6.1) results in a blowup precisely at those N points at t = T , and it takes the profile of the sharply scaled harmonic map (6.2) around each of the N points with the following blowup speed:…”

Trichotomy dynamics of the 1-equivariant harmonic map flow

“…where Γ 4 is the heat kernel in R 4 , and we have used the fact τ d ≫ R 2 ln R and the convolution estimates in [36,Lemma A.2]. The proof of (B.4) is complete.…”