AI 中文总结
该研究将OpenAI的Navier-Stokes爆破解构造推广到全空间，确定了光滑力在$L^1_tH^s_x$和$L^2_tH^s_x$拓扑中稠密的Sobolev阈值（$s<1/2$和$s<-1/2$），并证明了能量逼近与网格观测精确性。
AI 中文摘要
我们从OpenAI建立的紧支撑、光滑强迫的Navier-Stokes爆破解构造中推导出全空间分布结果。对于每个固定的正粘性和截止时间，当且仅当$s<1/2$时，在相对$L^1_tH^s_x$拓扑中，导致从静止状态在$\mathbb R^3$上经典破裂的光滑力是稠密的；当且仅当$s<-1/2$时，在相对$L^2_tH^s_x$拓扑中也是如此。正结果保留了$H^\infty_\sigma(\R^3)$中的每个固定初始速度。它们通过紧支撑无散度去除正则背景，然后精确插入单个全空间奇异包来实现。独立的临界估计证明了非稠密性；低频空间频率被显式控制，无需周期均值或Poincaré不等式。结论适用于时空紧支撑力和Clay公式中的快速衰减数据类。我们还获得了在通常的完备能量-力空间中的稠密性，正则轨迹的强能量和耗散逼近，以及在指定的有限全空间网格族上单元观测的精确相等性。初始状态空间、力空间和数值误差正则性在整个过程中被区分。没有声称固定力不稳定性、弱存在性丧失或新的形式验证。
英文摘要
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 $H^\infty_σ(\R^3)$. 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 Poincaré 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.
CommentsWaiting for formalization
