Lipschitz regularity of harmonic map heat flows into $CAT(0)$ spaces

Technical Summary: Lipschitz Regularity of Harmonic Map Heat Flows into CAT(0) Spaces

Problem Statement
The paper addresses the long-standing open question regarding the regularity of weak solutions to the harmonic map heat flow into CAT(0) spaces. While the existence, uniqueness, and long-time convergence of such weak solutions were established by Mayer and Jost in the 1990s via gradient flow theory in metric spaces, the question of whether these solutions possess Lipschitz regularity in both space and time remained unresolved. Previous work by Lin, Segatti, Sire, and Wang (2026) provided a partial answer, establishing Lipschitz continuity in the spatial variable for a specific class of CAT(0) spaces (those embeddable in Euclidean space with specific asymptotic properties) and 1/2-Hölder continuity in time. The authors aim to provide a complete answer for arbitrary CAT(0) spaces, proving Lipschitz regularity in both variables and establishing a parabolic Bochner-type inequality.

Methodology
The authors employ a combination of intrinsic geometric analysis, viscosity solution theory, and nonlinear flow techniques adapted from their previous work on harmonic maps. The proof strategy is divided into three main stages:

1. Local Hölder Continuity:
   The authors first establish local Hölder continuity for the weak solutions. They adapt the De Giorgi-Nash-Moser theory to the parabolic setting. A key difficulty is that the standard parabolic Poincaré inequality fails for general functions in $W^{1,2}$ due to the lack of time derivatives on the right-hand side. However, utilizing the regularity of the gradient flow (specifically the Lipschitz continuity of the flow in the $L^2$ metric for $t > 0$), they prove a modified local Poincaré inequality. This allows them to derive a subsolution property for the distance function $d_Y(P, u(x,t))$, leading to local Hölder continuity via iteration arguments.

2. Lipschitz Continuity in Time:
   To prove Lipschitz continuity in the time variable, the authors establish a crucial "subsolution property" for the squared distance between two weak solutions, $w(x,t) = d_Y^2(u(x,t), v(x,t))$. They prove that $(\Delta - \partial_t)w \geq 2R_{u,v}$ in the sense of distributions, where $R_{u,v}$ is a non-negative term related to the difference in energy densities. By choosing $v(x,t) = u(x, t+s)$ and applying the maximum principle to this subsolution, they derive a uniform bound on the time difference, proving that $t \mapsto u(x, \cdot)$ is locally Lipschitz continuous.

3. Lipschitz Continuity in Space:
   The spatial regularity is established by constructing a nonlinear "Hamilton-Jacobi" flow. Unlike the elliptic case where the flow involves two spatial variables, the parabolic case initially suggested a dependence on two time variables ($t$ and $s$), which complicated the derivation of a parabolic PDE. Leveraging the previously established time-Lipschitz continuity, the authors define a "revised" flow depending only on one time variable:
   $$f_\epsilon(x, t) := \inf_{y \in \Omega} \left( \frac{e^{-2Kt} d^2(x, y)}{2\epsilon} - d_Y(u(x, t), u(y, t)) \right)$$
   They prove that this function is a viscosity supersolution of the heat equation. By connecting this flow to the spatial gradient norm (via an asymptotic mean value inequality and properties of the Hardy-Littlewood maximal function), they bound the spatial Lipschitz constant, thereby establishing local Lipschitz continuity in space.

4. Bochner Inequality:
   Finally, the authors refine the Hamilton-Jacobi approach by lifting the power of the distance function from index 2 to a higher integer $p$. By analyzing the limit as $p \to \infty$, they derive an Eells-Sampson-type Bochner inequality for the spatial Lipschitz constant.

Key Results
The paper establishes the following main theorems:

• Theorem 1.3 (Main Regularity Result): Let $M$ be a complete Riemannian manifold with Ricci curvature bounded below by $K$, and let $(Y, d_Y)$ be an arbitrary CAT(0) space. For any weak solution $u(x,t)$ of the harmonic map heat flow with bounded initial data, $u$ is locally Lipschitz continuous on $\Omega \times (0, +\infty)$ in both space and time.
• Theorem 1.3 (ii) (Bochner Inequality): The pointwise spatial Lipschitz constant, defined as $\text{lip}_x u(x,t) := \limsup_{y \to x} \frac{d_Y(u(x,t), u(y,t))}{d(y,x)}$, belongs to $V_{2,loc} \cap L^\infty_{loc}$ and satisfies the distributional inequality:
  $$(\Delta - \partial_t)(\text{lip}_x u)^2 \geq 2|\nabla \text{lip}_x u|^2 + 2K(\text{lip}_x u)^2$$
  This is a parabolic analogue of the Bochner inequality for harmonic maps.

Significance and Claims
The authors claim to provide a "complete answer" to the question of Lipschitz regularity for harmonic map heat flows into singular metric spaces, removing the restrictive embedding assumptions required in previous partial results (Lin et al.). The results hold for any CAT(0) space, regardless of local compactness.

The paper emphasizes that the derived Bochner inequality (Theorem 1.3(ii)) is a significant tool, analogous to the classical Eells-Sampson inequality, which is central to rigidity theorems in geometric group theory. The authors note that their arguments in Sections 3, 4, and 5 can be adapted to maps from RCD spaces into CAT(0) spaces, though the spatial Lipschitz proof (Section 7) currently relies on viscosity solution theory which is not yet fully developed for parabolic equations on RCD spaces.

The work builds directly upon the foundational gradient flow theory of Mayer and Jost and the recent elliptic approximation methods of Lin, Segatti, Sire, and Wang, extending these frameworks to achieve full space-time regularity.