AI 中文总结
该研究基于凸积分格式，利用移动山涡旋构造三维环面上不可压缩纳维-斯托克斯方程的弱解，证明了特定指数下的非唯一性，为方程解的灵活性提供了新结果。
AI 中文摘要
我们在三维环面上构造了三维不可压缩纳维-斯托克斯方程的弱解，该凸积分格式基于Bruè、Colombo和Kumar的移动偶极子构造方法。对于显式指数\\(\bar p=\frac{6}{5}+5\times10^{-5}\\)，以及\\(L^2(\mathbb T^3)\\)中任意两个均值为零、散度为零的向量场，我们构造了一个弱解，其在时刻0和1的迹可任意逼近给定场，且满足\\(u\in C([0,1];L^2(\mathbb T^3))\\)，\\(\nabla u\in C([0,1];L^{\bar p}(\mathbb T^3))\\)。利用迭代的时间局部性，我们还得到了\\(L^2_\sigma(\mathbb T^3)\\)中稠密初始数据集的精确非唯一性。主要扰动是山球形涡旋的局部化、重缩放副本，山缩放既保持动能尺度，也保持速度梯度的\\(L^{6/5}\\)尺度。该构造利用了势外流的局部化、移动涡核的长轨道平均、辅助源校正以及与时间一致索伯列夫控制兼容的时间校正。
英文摘要
We construct weak solutions of the three-dimensional incompressible Navier--Stokes equations on the torus. The convex-integration scheme is based on the moving-dipole construction of Bruè, Colombo, and Kumar~\cite{BrueColomboKumar2024}. For the explicit exponent $\bar p=\frac65+5\times10^{-5},$ and for any two mean-zero, divergence-free vector fields in $L^2(\mathbb T^3)$, we construct a weak solution whose traces at times $0$ and $1$ approximate the prescribed fields arbitrarily well and which satisfies \[ u\in C([0,1];L^2(\mathbb T^3)), \qquad \nabla u\in C([0,1];L^{\bar p}(\mathbb T^3)). \] Exploiting the time-locality of the iteration, we also obtain exact nonuniqueness for a dense set of initial data in \(L^2_σ(\mathbb T^3)\).The principal perturbations are localized, rescaled copies of Hill's spherical vortex. The Hill scaling preserves both the kinetic-energy scale and the \(L^{6/5}\)-scale of the velocity gradient. The construction uses localization of the potential exterior flow, long-orbit averaging of moving vortex cores, an auxiliary source correction, and a temporal corrector compatible with uniform-in-time Sobolev control.
