Blowup dynamics for smooth equivariant solutions to energy critical Landau-Lifschitz flow

TL;DR

This paper investigates finite-time blowup solutions in the energy critical Landau-Lifschitz flow on  to , revealing a codimension-one set of initial data leading to singularity formation with a rate independent of coefficients.

Contribution

It constructs and characterizes type-II blowup solutions near the ground state for the energy critical Landau-Lifschitz flow, including a detailed description of singularity formation.

Findings

Existence of a codimension-one set of initial data causing blowup.

Precise description of the blowup singularity.

Blowup rate is independent of the flow coefficients.

Abstract

In this paper, we study the energy critical 1-equivariant Landau-Lifschitz flow mapping $\mathbb{R}^2$ to $\mathbb{S}^2$ with arbitrary given coefficients $\rho_1\in \mathbb{R}$, $\rho_2>0$. We prove that there exists a codimension one smooth well-localized set of initial data arbitrarily close to the ground state which generates type-II finite-time blowup solutions, and give a precise description of the corresponding singularity formation. In our proof, both the Schr\"odinger part and the heat part play important roles in the construction of approximate solutions and the mixed energy/Morawetz functional. However, the blowup rate is independent of the coefficients.