Failure of analyticity-radius growth in energy-canceling fluid models
Abstract
We prove that exact quadratic energy cancellation alone does not force growth of the spatial analyticity radius. On 𝕋3, we construct an explicit symmetric, translation-invariant, first-order bilinear operator Q that preserves the divergence-free class and, for every smooth real-valued divergence-free vector field v, satisfies $ \int_{\mathbb{T}^3} Q(v,v)\cdot v\,dx=0.$ For every prescribed sufficiently small time T>0, the equation ∂t u−Δu=Q(u,u) admits a global smooth, real-valued, mean-zero, divergence-free solution u such that rad(u(0))=rad(u(T))=1. The construction reduces the dynamics on an invariant cyclic-shear class to viscous Burgers and tunes a Cole-Hopf heat profile so that its nearest complex zero returns to its initial distance from the real torus. For every 1<α<2 and every prescribed sufficiently small T>0, we also construct a symmetric sparse frequency set, its associated Fourier projection, and trigonometric-polynomial initial data for the projected dissipative surface quasi-geostrophic equation. The resulting unique global smooth solution θ has infinite analyticity radius initially but satisfies 0<rad(θ(T))≤1. An additively separated Fourier cascade yields coefficientwise exponential lower bounds, while uniform comparison estimates control the feedback interactions. Thus entire analyticity need not persist even from trigonometric-polynomial data. Together, the two constructions show that an exact energy identity alone does not determine the frequency geometry governing analyticity-radius growth.
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BibTeX
@misc{chen2026failure,
title = {{Failure of analyticity-radius growth in energy-canceling fluid models}},
author = {Ke Chen and Haina Li and Quoc-Hung Nguyen and Ping Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05023},
eprint = {2609.05023},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05023},
}
RIS
TY - UNPB
TI - Failure of analyticity-radius growth in energy-canceling fluid models
AU - Ke Chen
AU - Haina Li
AU - Quoc-Hung Nguyen
AU - Ping Zhang
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05023
UR - https://arxiv.org/abs/2609.05023
AB - We prove that exact quadratic energy cancellation alone does not force growth of the spatial analyticity radius. On 𝕋^3, we construct an explicit symmetric, translation-invariant, first-order bilinear operator Q that preserves the divergence-free class and, for every smooth real-valued divergence-free vector field v, satisfies $ \int_{\mathbb{T}^3} Q(v,v)\cdot v\,dx=0.$ For every prescribed sufficiently small time T>0, the equation ∂_t u−Δu=Q(u,u) admits a global smooth, real-valued, mean-zero, divergence-free solution u such that rad(u(0))=rad(u(T))=1. The construction reduces the dynamics on an invariant cyclic-shear class to viscous Burgers and tunes a Cole-Hopf heat profile so that its nearest complex zero returns to its initial distance from the real torus. For every 1<α<2 and every prescribed sufficiently small T>0, we also construct a symmetric sparse frequency set, its associated Fourier projection, and trigonometric-polynomial initial data for the projected dissipative surface quasi-geostrophic equation. The resulting unique global smooth solution θ has infinite analyticity radius initially but satisfies 0<rad(θ(T))≤1. An additively separated Fourier cascade yields coefficientwise exponential lower bounds, while uniform comparison estimates control the feedback interactions. Thus entire analyticity need not persist even from trigonometric-polynomial data. Together, the two constructions show that an exact energy identity alone does not determine the frequency geometry governing analyticity-radius growth.
ER -