Global dissipative half-harmonic flows into spheres: small data in critical Sobolev spaces

TL;DR

This paper proves the global existence, uniqueness, and long-term behavior of solutions to certain dissipative geometric flows, specifically the half-harmonic heat flow and Landau-Lifshitz equation, for small initial data in critical Sobolev spaces.

Contribution

It establishes the well-posedness and asymptotic properties of these flows in low dimensions for small initial data in critical Sobolev norms, extending understanding of their long-time dynamics.

Findings

Global solutions exist and are unique for small initial data.

Solutions exhibit regularity and specific long-time asymptotic behavior.

Results apply to space dimensions up to three.

Abstract

We establish global existence, uniqueness, regularity and long-time asymptotics of strong solutions to the half-harmonic heat flow and dissipative Landau-Lifshitz equation, valid for initial data that is small in the homogeneous Sobolev norm $\dot{H}^{\frac n 2}$ for space dimensions $n \le 3$.