AI 中文总结
该研究证明三维欧拉与纳维-斯托克斯方程单壳解的单粒子变形梯度可实现整个\boldsymbol{\rm SL}(3,\boldsymbol{\rm R})群，分类了确定无迹应变的有限材料方向系统，明确了鲁棒应变传感的精确标量通道数。
AI 中文摘要
我们证明了，三维欧拉方程和纳维-斯托克斯方程的周期性、无外力、单壳解的单粒子变形梯度构成整个\boldsymbol{\rm SL}(3,\boldsymbol{\rm R})群。更准确地说，给定粒子标签\boldsymbol{p}\boldsymbol{\rm T}^3、时间\boldsymbol{T}>0和\boldsymbol{F}_*\boldsymbol{\rm SL}(3,\boldsymbol{\rm R})，每个足够大的奇数\boldsymbol{N}都存在实解析旋量本征场\boldsymbol{W}_N}满足\boldsymbol{\rm curl}\boldsymbol{W}_N=N\boldsymbol{W}_N}。结合显式标量振幅，\boldsymbol{W}_N}可得到稳态欧拉解；结合显式指数衰减振幅，对任意正粘度可得到纳维-斯托克斯解，两种情形下均满足\boldsymbol{\nabla}_a\boldsymbol{X}(\boldsymbol{p},\boldsymbol{T})=\boldsymbol{F}_*}。提升后的粒子轨迹是速度处处非零的嵌入解析弧。该构造结合了对\boldsymbol{\rm SL}(3,\boldsymbol{\rm R})的全局无迹对称矩阵控制、沿受控弧的贝尔特拉米柯西问题、龙格逼近、环面上的逆局域化以及有限维端点修正。我们还对有限材料方向系统进行分类，该系统在任意保体积变形后确定所有无迹应变：在\boldsymbol{n}维空间中，当相关秩一算子张成\boldsymbol{\rm Sym}(n)}时，此同余鲁棒性质恰好成立，因此标量通道的精确数量为\boldsymbol{n(n+1)/2}；在三维空间中，最小系统中未变形外积下界的最大值恰好由正二十面体的六个轴实现。该实现定理表明，此传感结果中的全变形群量化器可在上述刚性类内动态达到。
英文摘要
We prove that the full group \(\SL(3,\R)\) occurs as the set of one-particle deformation gradients of periodic, unforced, single-shell solutions of both the three-dimensional Euler and Navier--Stokes equations. More precisely, given a particle label \(p\in\T^3\), a time \(T>0\), and \(F_*\in\SL(3,\R)\), every sufficiently large odd integer \(N\) admits a real-analytic curl eigenfield \(W_N\) with \(\operatorname{curl}W_N=NW_N\). With an explicit scalar amplitude, \(W_N\) yields a steady Euler solution; with an explicit exponentially decaying amplitude, it yields a Navier--Stokes solution for any positive viscosity, and in both cases \(\nabla_aX(p,T)=F_*\). The lifted particle trajectory is an embedded analytic arc with nowhere-vanishing velocity. The construction combines global trace-free symmetric-matrix control on \(\SL(3,\R)\), a Beltrami Cauchy problem along the controlled arc, Runge approximation, inverse localization on the torus, and a finite-dimensional endpoint correction. We also classify finite material-direction systems that determine every trace-free strain after arbitrary volume-preserving deformation. In dimension \(n\), this congruence-robust property holds exactly when the associated rank-one projectors span \(\Sym(n)\); hence the sharp number of scalar channels is \(n(n+1)/2\). In dimension three, among minimal systems the undeformed outer-product lower bound is maximized exactly by the six axes of a regular icosahedron. The realization theorem shows that the full deformation-group quantifier in this sensing result is dynamically attained within the rigid class above.
