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

Juncheng Wei, Qidi Zhang, Yifu Zhou

TL;DR

This work proves finite-time blow-up for the Landau-Lifshitz-Gilbert equation in 2D with target $S^2$ by constructing multi-bubble blow-up solutions at prescribed points. A parabolic inner-outer gluing framework is developed, combining a distorted Fourier transform for the inner problem (notably mode $-1$) with a $\mathsf{DMO_x}$-based regularity theory for the outer parabolic system, to handle the dispersive-damped nature of LLG. The main theorem yields type II blow-up with sharp profiles around each blow-up point and a precise rate $\lambda_j(t) \sim \kappa_j^*\lambda_*(t)$ where $\lambda_*(t)=\tfrac{|\ln T|(T-t)}{|\ln(T-t)|^2}$ and $\|\nabla u\|_{L^{\infty}} \sim \tfrac{|\ln(T-t)|^2}{T-t}$ as $t\to T$, while the gradient measures converge to a sum of $8\pi\delta$-masses at the blow-up points. The construction relies on carefully balancing inner and outer corrections, nonlocal reduced equations for the bubble parameters, and fixed-point arguments in weighted function spaces, providing a robust template for non-equivariant bubbling in dispersive parabolic systems. The results illuminate the interaction of damping and dispersion in singularity formation and extend the parabolic gluing methodology to quasilinear, dispersive contexts, with potential extensions to bounded domains and stability analyses.

Abstract

We construct finite time blow-up solutions to the Landau-Lifshitz-Gilbert equation (LLG) from ${\mathbb R}^2$ into $S^2$ \begin{equation*} \begin{cases} u_t= a(Δu+|\nabla u|^2u) -b u\wedge Δu &\ \mbox{ in }\ {\mathbb R}^2\times(0,T), u(\cdot,0) = u_0\in S^2 &\ \mbox{ in }\ {\mathbb R}^2, \end{cases} \end{equation*} where $a^2+b^2=1,~a > 0,~ b\in {\mathbb R}$. Given any prescribed $N$ points in $\mathbb{R}^2$ 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 speed \begin{equation*} \| \nabla u\|_{L^\infty } \sim \frac{|\ln(T-t)|^2}{ T-t } \ \mbox{ as } \ t\to T. \end{equation*} The proof is based on the {\em parabolic inner-outer gluing method}, developed in \cite{17HMF} 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 \cite{Krieger09Duke,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 ($\mathsf{DMO_x}$), which was proved by Dong, Kim and Lee \cite{dong22-non-divergence} recently.

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

TL;DR

This work proves finite-time blow-up for the Landau-Lifshitz-Gilbert equation in 2D with target by constructing multi-bubble blow-up solutions at prescribed points. A parabolic inner-outer gluing framework is developed, combining a distorted Fourier transform for the inner problem (notably mode ) with a -based regularity theory for the outer parabolic system, to handle the dispersive-damped nature of LLG. The main theorem yields type II blow-up with sharp profiles around each blow-up point and a precise rate where and as , while the gradient measures converge to a sum of -masses at the blow-up points. The construction relies on carefully balancing inner and outer corrections, nonlocal reduced equations for the bubble parameters, and fixed-point arguments in weighted function spaces, providing a robust template for non-equivariant bubbling in dispersive parabolic systems. The results illuminate the interaction of damping and dispersion in singularity formation and extend the parabolic gluing methodology to quasilinear, dispersive contexts, with potential extensions to bounded domains and stability analyses.

Abstract

We construct finite time blow-up solutions to the Landau-Lifshitz-Gilbert equation (LLG) from into \begin{equation*} \begin{cases} u_t= a(Δu+|\nabla u|^2u) -b u\wedge Δu &\ \mbox{ in }\ {\mathbb R}^2\times(0,T), u(\cdot,0) = u_0\in S^2 &\ \mbox{ in }\ {\mathbb R}^2, \end{cases} \end{equation*} where . Given any prescribed points in and small , we prove that there exists regular initial data such that the solution blows up precisely at these points at finite time , taking around each point the profile of sharply scaled degree 1 harmonic map with the type II blow-up speed \begin{equation*} \| \nabla u\|_{L^\infty } \sim \frac{|\ln(T-t)|^2}{ T-t } \ \mbox{ as } \ t\to T. \end{equation*} The proof is based on the {\em parabolic inner-outer gluing method}, developed in \cite{17HMF} 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 \cite{Krieger09Duke,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 (), which was proved by Dong, Kim and Lee \cite{dong22-non-divergence} recently.
Paper Structure (50 sections, 34 theorems, 945 equations)

This paper contains 50 sections, 34 theorems, 945 equations.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Main results
  4. Strategy and novelties in the construction
  5. Distorted Fourier transform and re-gluing process in the inner problems
  6. Regularity estimates in the $\mathsf{DMO_x}$-class for the outer problem
  7. Improvement of slow decay and tricks used in the interacting error terms
  8. Comments on other related problems and techniques
  9. Main steps of the construction
  10. Notations and preliminaries
  11. Approximation and improvement
  12. First approximation
  13. Global corrections by parabolic systems
  14. The upper bounds of global correction terms
  15. New errors produced by the global corrections
  16. ...and 35 more sections

Key Result

Theorem 1

Assume $a^2+b^2=1$, $a>0$, $b\in \mathbb{R}$ in LLG-eq. Given $N\in \mathbb Z_+$ and arbitrary $N$ distinct points $q^{ {\hbox{\tiny{$[j]$}} } }\in \mathbb R^2$, $j=1,2,\dots, N$, for $T>0$ sufficiently small, there exists a smooth initial data $u_0$ such that the gradient of the solution $u$ to LLG with where $\kappa_j^*>0$ is a constant independent of $a,b$, $\gamma_j^*\in [-\pi/2, \pi/2]$ is a

Theorems & Definitions (83)

  • Theorem 1
  • Corollary 1.1
  • Remark 1.1
  • proof : Proof of Claim \ref{['nonlocal-est']}
  • Lemma 5.1
  • proof
  • Proposition 5.1
  • Lemma 6.1
  • proof
  • Proposition 6.1
  • ...and 73 more