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

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 Schrödinger 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.

84 pages

References in corpus (5)

• Stable blow up dynamics for the 1-corotational energy critical harmonic heat flow
• Equivariant Schrödinger Maps in two spatial dimensions
• Global Schrödinger map flows to Kähler manifolds with small data in critical Sobolev spaces: Energy critical case
• Global Schrödinger map flows to Kähler manifolds with small data in critical Sobolev spaces: High dimensions
• Asymptotic behaviors of Landau-Lifshitz flows from ${\mathbb{R}}^2$ to Kähler manifolds