arXiv subjects

Zhuoni Chi

Publications and source records attributed to Zhuoni Chi.

3 recordsLinked to original sources

arXiv2026

Distribution of Singular Data Generated by Compact Forced Navier-Stokes Blowup

Starting from the compact, smoothly forced Navier--Stokes blowup solution constructed by OpenAI, we study the distribution of the singular data it generates. For fixed viscosity and time horizon on the three-dimensional torus, smooth forces producing classical breakdown from rest by that time are dense in the inherited time-integrated spatial $H^s$ topology if and only if $s<1/2$ (the norms are defined in Section~\ref{sec:intro}). The positive result follows from an exact insertion into every regular reference trajectory. A localized vector potential removes the background around a concentrated singular packet, so all nonlinear cross terms vanish and the modified force remains smooth through the singular time. We give the full support construction, derivative estimates, fractional Sobolev scaling and classical-lifespan argument. The inserted trajectories converge strongly in the energy and dissipation norm. A separate critical-force bootstrap, followed by an $H^1$ estimate and high-Sobolev continuation, supplies the regular open set needed for non-density at and above $s=1/2$. Further results describe extended-data projections, every fixed smooth initial-velocity slice, mixed force norms, infinite-dimensional variations and interior no-slip realizations. The construction varies the force; it does not classify singular initial velocities for a single prescribed force.

math.AP↗

arXiv2026

Singular Forces on the Whole Space: Sobolev Density Thresholds and Energy Approximation

We derive whole-space distribution results from the established compact, smoothly forced Navier--Stokes blowup construction of OpenAI. For every fixed positive viscosity and deadline, smooth forces causing classical breakdown from rest on $\mathbb R^3$ are dense in the relative $L^1_tH^s_x$ topology exactly when $s<1/2$, and in the relative $L^2_tH^s_x$ topology exactly when $s<-1/2$. The positive results preserve every fixed initial velocity in $H^\infty_\sigma(\R^3)$. They follow from a compact divergence-free removal of a regular background, followed by exact insertion of a single whole-space singular packet. Separate critical estimates prove non-density; low spatial frequencies are controlled explicitly, without a periodic mean or a Poincar\'e inequality. The conclusions hold for spacetime-compact forces and for the rapidly decaying data class in the Clay formulation. We also obtain density in the usual completed energy-force spaces, strong energy-and-dissipation approximation of regular trajectories, and exact equality of cell observations on a prescribed finite family of whole-space grids. Initial-state spaces, forcing spaces and numerical error regularity are distinguished throughout. No claim of fixed-force instability, loss of weak existence or new formal verification is made.

math.AP↗

arXiv2026

Spherical representations of unitary groups at ramified places and the arithmetic inner product formula

In this article, we study admissible representations of even unitary groups over local fields, where the quadratic extension is ramified, with invariant vectors under the action of the stabilizer of a unimodular lattice and some properties of the corresponding integral model of unitary Shimura varieties. As a direct application, we are able to improve the arithmetic inner product formula so that the places with local root number $(-1)$ are allowed to be ramified.

math.NT↗