Remarks on the heat flow of harmonic maps into CAT(0)-spaces

In this paper, we present an alternate, elementary proof of the local Lipschitz regularity of the suitable weak solution of heat flow of harmonic maps into CAT(0)-metric spaces, whose existence was established by Lin, Segatti, Sire, and Wang through an elliptic regularization approach. The ideas of the proof are inspired by Korevaar and Schoen, and they work for any CAT(0)-metric space $(X,d)$ as the target and any complete Riemanan manifold $(M,g)$, with positive injectivity radius and bounded curvature, as the domain.

💡 Research Summary

This paper provides a new, elementary proof of the local Lipschitz regularity for suitable weak solutions of the heat flow of harmonic maps into CAT(0) metric spaces. The existence of such solutions was previously established by Lin, Segatti, Sire, and Wang via an elliptic regularization (Weighted Energy Dissipation, WED) method. The authors’ approach is inspired by the comparison techniques of Korevaar–Schoen and works for any CAT(0) target space (X,d) and any complete Riemannian domain (M,g) with positive injectivity radius and bounded curvature.

The core of the argument is a two‑step observation. First, any suitable weak solution satisfies the Evolution Variational Inequality (EVI). By testing the EVI with a smooth non‑negative cutoff η, the authors derive a parabolic inequality (1.6) that couples the spatial gradient |∇u|² with the time derivative |∂ₜu|². Second, they recall that |∂ₜu|² is a sub‑solution of the heat operator, i.e. (∂ₜ‑2Δ)|∂ₜu|² ≥ 0 in the distributional sense. Combining these two facts yields a nonlinear parabolic inequality for |∇u|². This inequality is precisely of the type to which Moser’s iteration and the classical Harnack inequality apply. Consequently, |∇u|² is locally bounded, which implies that u is locally Lipschitz in the spatial variables for any positive time. This result is stated as Theorem 1.3 and holds on any complete Riemannian manifold with the stated geometric bounds.

In addition to the abstract regularity result, the paper establishes a uniform Lipschitz estimate for the ε‑regularized approximations u_ε obtained from the WED functional. In the Euclidean setting (M=ℝⁿ), the authors use a translation argument: for a small vector e they compare u_ε(x+e,t) with u_ε(x,t) and consider the finite difference quotient of the squared distance. By exploiting the CAT(0) quadrilateral inequality (Reshetnyak’s comparison) they show that the difference quotient satisfies a weak sub‑solution property. Passing to the limit as the translation step h→0 yields the inequality (ε∂ₜ²‑∂ₜ+Δ)|∇u_ε|² ≥ 0. Together with the energy bound for u_ε, this leads to the uniform estimate (Theorem 1.4): for any parabolic cylinder B_r(x₀)×(t₀‑r²,t₀+r²) and any ε≤r², \