Short-time 1/√t regularization for EVI flows in metric spaces

Characterize the metric spaces and energy functionals E for which EVI_λ gradient flows exhibit the stronger short-time regularization bound |∂E|(γ_t) ≲ t^{-1/2} as t→0+, extending the known Wasserstein-space cases to a general metric space framework.

Background

In Wasserstein settings, the authors establish a short-time bound |∂E|(ρ_t) ≤ C t{-1/2} via Fisher-information estimates (e.g., shift-Harnack or Li–Yau inequalities) to ensure finite length for concatenated flow curves.

They note that generic EVI tools only yield a weaker O(1/t) bound, insufficient for integrability, motivating the problem of identifying structural conditions under which the sharper O(t{-1/2}) rate holds in abstract metric spaces.

References

This points to an interesting open problem, which is to characterize when this stronger regularization rate holds for EVI flows in the metric space setting.

— Geodesic convexity and strengthened functional inequalities in submanifolds of Wasserstein space  (2508.13698 - Chaintron et al., 19 Aug 2025) in Section 3.3 (discussion following Proposition 3.2)