A clarification on the distinction between nonlocal error and profile residual error in a computer-assisted proof of 3D Euler singularity

Abstract

In \cite{zhang2025dimension}, the author discusses the applicability of a nonexistence result for self-similar profiles to the Hou--Luo scenario and raises questions about the approximate self-similar profile constructed in \cite{ChenHou2023a,ChenHou2023b}. In this note, we clarify two main points. First, the far-field discrepancy identified in \cite{zhang2025dimension} is the raw nonlocal Poisson error, not the profile residual error used in the stability proof of \cite{ChenHou2023a,ChenHou2023b}. The relevant nonlocal contribution to the profile residual contains the nonlocal velocity error multiplied by additional decaying factors, which provide crucial smallness in the far field estimates. Second, we examine the figure-based evidence used in \cite[Remark 2.4]{zhang2025dimension} to support the applicability of the assumption in \cite[Proposition 2.3(i)]{zhang2025dimension}, and show that the same grid-point data do not support that interpretation. Thus the comparisons and figure-based evidence in \cite{zhang2025dimension} neither invalidate the residual estimates in \cite{ChenHou2023a,ChenHou2023b} nor justify the claimed applicability of the angular-increase assumption to the Hou--Luo scenario.

7 pages