Diffeomorphism Invariance of the Favre-Filtered Compressible Navier--Stokes System under Spacetime Dilation

RESUMO

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 $\{\phi_L\}_{L>0}$, verify that each $\phi_L$ 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 $\phi_L$ 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 $L$.