Mathematical Framework Resolves Viscosity Operator Ambiguity in Navier-Stokes Equations on Curved Spaces

The Navier-Stokes equations, which govern viscous fluid flow, have multiple inequivalent formulations on curved (Riemannian) manifolds depending on which viscous operator is chosen: the Hodge Laplacian, the Bochner Laplacian, or the deformation Laplacian. A new preprint by Zhi-Wei Wang argues that a Lagrangian kinematic construction — building the strain rate from the rate of change of inner products of Lie-dragged connecting vectors — uniquely selects the deformation Laplacian before any constitutive assumptions are made, excluding the Hodge Laplacian on purely kinematic grounds. The paper also analyzes the thin-shell limit of three-dimensional flows, showing that stress-free (Navier slip) boundary conditions recover the deformation Laplacian while Hodge boundary conditions recover the Hodge Laplacian, with the intrinsic component always being the deformation Laplacian regardless of boundary choice. As analytical confirmation, the authors note that the deformation Laplacian is coercive on the hyperbolic plane while the Hodge Laplacian is not, consistent with the known failure of the energy inequality for the latter. The paper concludes by proving that on any complete two-dimensional manifold with Gaussian curvature bounded above by a negative constant, the incompressible Navier-Stokes equation with the deformation Laplacian admits a unique global weak solution with exponential energy decay — a result that was previously obstructed for the Hodge Laplacian formulation. The preprint is 15 pages and has been submitted to arXiv under mathematical physics, analysis of PDEs, differential geometry, and fluid dynamics.