AI 中文总结
本文证明Favre滤波可压缩Navier-Stokes方程组在时空膨胀下具有微分同胚不变性，通过构造膨胀群并验证各方程残差同乘因子，得出所有参数解空间同构。
AI 中文摘要
我们证明，在任意参数$L>0$的坐标$(Lt,Lx_1,Lx_2,Lx_3)$下写出的、带有亚格子尺度闭合项的Favre滤波可压缩Navier--Stokes方程组，可以通过底层时空流形上的光滑微分同胚作用，从标准坐标$(t,x_1,x_2,x_3)$下写出的方程组获得。证明始于对无限维流形和光滑场构型的Fréchet流形的自包含处理。随后我们定义单参数膨胀群$\{\phi_L\}_{L>0}$，验证每个$\phi_L$是微分同胚，并计算其切与余切作用。我们证明，诱导的构型丛截面的拉回作用与全导数算子交换，这意味着$\phi_L$的一阶射流延拓将方程子流形映射到自身。逐方程验证——连续性、动量与能量——确认每个残差都乘以公共非零因子$L^{-1}$（或等价地，在变换坐标下恒为零），从而解集得以保持。我们得出结论：由$L>0$参数化的方程组族满足微分同胚变换表达式，且所有$L$的解空间典范同构。
英文摘要
We prove that the system of Favre-filtered compressible Navier--Stokes equations with subgrid-scale closure terms, written in the coordinates $(Lt,Lx_1,Lx_2,Lx_3)$ for an arbitrary parameter $L>0$, is obtained from the system written in standard coordinates $(t,x_1,x_2,x_3)$ by the action of a smooth diffeomorphism of the underlying spacetime manifold. The proof begins with a self-contained treatment of infinite-dimensional manifolds and the Fréchet manifold of smooth field configurations. We then define the one-parameter dilation group $\{ϕ_L\}_{L>0}$, verify that each $ϕ_L$ is a diffeomorphism, and compute its tangent and cotangent actions. The induced pullback action on sections of the configuration bundle is shown to commute with the total derivative operator, which implies that the first-jet prolongation of $ϕ_L$ maps the equation submanifold of the system onto itself. Equation-by-equation verification---continuity, momentum, and energy---confirms that every residual is multiplied by the common nonzero factor $L^{-1}$ (or, equivalently, is identically zero in the transformed coordinates), so that the solution set is preserved. We conclude that the family of systems parameterised by $L>0$ satisfies the diffeomorphism transformation expression and that solution spaces are canonically isomorphic for all $L$.
Comments14 pages
