Failure of analyticity-radius growth in energy-canceling fluid models
We prove that exact quadratic energy cancellation alone does not force growth of the spatial analyticity radius. On $\mathbb{T}^3$, 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 $\partial_t u-\Delta u=Q(u,u)$ admits a global smooth, real-valued, mean-zero, divergence-free solution $u$ such that $\operatorname{rad}(u(0))=\operatorname{rad}(u(T))=1$. 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 $1<\alpha<2$ 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 $\theta$ has infinite analyticity radius initially but satisfies $0<\operatorname{rad}(\theta(T))\leq1$. 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.