[Paper Review] On uniqueness of heat flow of harmonic maps

Tao Huang, Changyou Wang|arXiv (Cornell University)|Aug 7, 2012

Geometric Analysis and Curvature Flows24 references3 citations

TL;DR

This paper establishes the uniqueness of weak solutions to the heat flow of harmonic maps into compact Riemannian manifolds, specifically the unit sphere $S^{k-1}$ or compact Riemannian homogeneous manifolds, under small renormalized energy conditions. Using Morrey space estimates and an $\epsilon_0$-regularity theorem, it proves that solutions with sufficiently small gradient norms in Morrey spaces must be identical, extending uniqueness results beyond the $n=2$ case and confirming a conjecture of Struwe for a broad class of initial data.

ABSTRACT

In this paper, we establish the uniqueness of heat flow of harmonic maps into (N, h) that have sufficiently small renormalized energies, provided that N is either a unit sphere $S^{k-1}$ or a compact Riemannian homogeneous manifold without boundary. For such a class of solutions, we also establish the convexity property of the Dirichlet energy for $t\ge t_0>0$ and the unique limit property at time infinity. As a corollary, the uniqueness is shown for heat flow of harmonic maps into any compact Riemannian manifold N without boundary whose gradients belong to $L^q_t L^l_x$ for $q>2$ and $l>n$ satisfying the Serrin's condition.

Motivation & Objective

• To resolve the long-standing open problem of uniqueness for weak solutions to the heat flow of harmonic maps in dimensions $n \geq 2$.
• To address Struwe's conjecture on the existence of a function class ensuring uniqueness of solutions.
• To establish uniqueness under small renormalized energy conditions for harmonic map heat flow into $S^{k-1}$ or compact Riemannian homogeneous manifolds.
• To prove the convexity of Dirichlet energy and the unique limit property at infinity for such solutions.
• To extend uniqueness results to general compact Riemannian manifolds via integrability conditions on gradients.

Proposed method

• Introduce Morrey space norms $M^{p,p}_{R_0}$ and $M^{p,2p}_{R_0}$ to quantify smallness of gradients and time derivatives.
• Establish an $\epsilon_0$-regularity theorem for harmonic maps in Morrey spaces, ensuring smoothness under small energy conditions.
• Use the small energy regularity result to derive a convexity inequality for Dirichlet energy in time $t \geq t_0 > 0$, implying uniqueness.
• Apply the Poincare9 and Hardy inequalities to control the difference between two solutions in terms of their gradient norms.
• Prove a comparison inequality between Dirichlet energies of two weak harmonic maps in a domain, showing that small gradient norms in Morrey space force identical solutions.
• Use the uniqueness result for harmonic maps in Morrey spaces to deduce uniqueness for the heat flow via energy comparison and time-regularity.

Experimental results

Research questions

• RQ1 Does the heat flow of harmonic maps into $S^{k-1}$ or compact Riemannian homogeneous manifolds admit unique weak solutions under small Morrey norm conditions on gradients and time derivatives?
• RQ2 Can the convexity of Dirichlet energy for $t \geq t_0 > 0$ be established for such solutions, implying long-time uniqueness?
• RQ3 Is Struwe's conjecture on uniqueness within a class defined by the monotonicity formula valid for $n \geq 2$ under small energy conditions?
• RQ4 Can the uniqueness result be extended to general compact Riemannian manifolds $N$ without boundary via integrability conditions on $\nabla u$?
• RQ5 What is the precise threshold of smallness (in terms of Morrey norms) that guarantees uniqueness of weak solutions to the heat flow?

Key findings

• For $n \geq 2$, if two weak solutions $u_1, u_2$ to the harmonic map heat flow satisfy $\max_{i=1,2} \left[ \|\nabla u_i\|_{M^{p,p}_{R_0}} + \|\partial_t u_i\|_{M^{p,2p}_{R_0}} \right] \leq \epsilon_0$, then $u_1 \equiv u_2$ on $M \times [0,T]$.
• The Dirichlet energy of the solution is convex in time for $t \geq t_0 > 0$ under the same smallness condition.
• The solution converges to a unique limit at infinity, i.e., $\lim_{t \to \infty} u(t)$ exists and is unique.
• For $N = S^{k-1}$ or a compact Riemannian homogeneous manifold, the uniqueness holds if $\|\nabla u_i\|_{M^{p,p}_{R_p}} \leq \epsilon_p$ for $1 < p \leq 2$, with $\epsilon_p$ and $R_p$ depending on $p$ and $\delta$.
• The result extends to general compact Riemannian manifolds $N$ without boundary if $\nabla u \in L^q_t L^l_x$ with $q > 2$ and $l > n$, satisfying condition (1.13).
• The proof relies on a new $\epsilon_0$-regularity theorem in Morrey spaces and a sharp convexity inequality for Dirichlet energy, derived via Poincare9 and Hardy inequalities.