A clarification on the distinction between nonlocal error and profile residual error in a computer-assisted proof of 3D Euler singularity
Jiajie Chen, Thomas Y. Hou
Abstract
In 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 ChenHou2023a,ChenHou2023b. In this note, we clarify two main points. First, the far-field discrepancy identified in zhang2025dimension is the raw nonlocal Poisson error, not the profile residual error used in the stability proof of 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 [Remark 2.4]zhang2025dimension to support the applicability of the assumption in [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 zhang2025dimension neither invalidate the residual estimates in ChenHou2023a,ChenHou2023b nor justify the claimed applicability of the angular-increase assumption to the Hou--Luo scenario.
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