Regularity and uniqueness of the heat flow of biharmonic maps

TL;DR

This paper proves regularity and uniqueness of the heat flow of biharmonic maps into spheres and general manifolds under energy smallness and integrability conditions, advancing understanding of their long-term behavior.

Contribution

It establishes regularity and uniqueness results for the heat flow of biharmonic maps into spheres and general manifolds, under specific energy and integrability conditions.

Findings

Regularity of heat flow under small energy conditions.

Uniqueness and convexity properties of solutions.

Existence of unique limits at infinite time.

Abstract

In this paper, we first establish regularity of the heat flow of biharmonic maps into the unit sphere $S^L\subset\mathbb R^{L+1}$ under a smallness condition of renormalized total energy. For the class of such solutions to the heat flow of biharmonic maps, we prove the properties of uniqueness, convexity of hessian energy, and unique limit at time infinity. We establish both regularity and uniqueness for the class of weak solutions $u$ to the heat flow of biharmonic maps into any compact Riemannian manifold $N$ without boundary such that $\nabla^2 u\in L^q_tL^p_x$ for some $p>n/2$ and $q>2$ satisfying (1.13).