Diffusive stability and self-similar decay for the harmonic map heat flow

TL;DR

This paper investigates the harmonic map heat flow on Euclidean space, establishing uniqueness, decay rates, and self-similar decay under certain initial conditions, with extensions to related flows and equations.

Contribution

It provides the first unconditional uniqueness results for small initial data in specific Besov spaces and characterizes self-similar decay for the harmonic map heat flow.

Findings

Unconditional uniqueness for small initial data in Besov spaces.

Decay rate of the gradient norm as t^{-1/2}.

Self-similar decay under localized initial conditions.

Abstract

In this paper we study the harmonic map heat flow on the euclidean space $\mathbb{R}^d$ and we show an unconditional uniqueness result for maps with small initial data in the homogeneous Besov space $\dot{B}^{\frac{d}{p}}_{p,\infty}(\mathbb{R}^d)$ where $d<p<\infty$. As a consequence we obtain decay rates for solutions of the harmonic map flow of the form $\|\nabla u(t) \|_{L^\infty(\mathbb{R}^d)}\leq Ct^{-\frac12}$. Additionally, under the assumption of a stronger spatial localization of the initial conditions, we show that the temporal decay happens in a self-similar way. We also explain that similar results hold for the biharmonic map heat flow and the semilinear heat equation with a power-type nonlinearity.