Resolving the viscosity operator ambiguity on Riemannian manifolds via a kinematic selection principle

Abstract

On a general Riemannian manifold the Navier-Stokes equations admit several inequivalent formulations, differing in the choice of viscous operator: the Hodge Laplacian, the Bochner Laplacian, or the deformation Laplacian. We show that a Lagrangian kinematic construction, in which the strain rate is built from the rate of change of inner products of Lie-dragged connecting vectors, uniquely selects the deformation Laplacian for fluids whose configuration space is intrinsically the manifold. The Hodge Laplacian is excluded at the kinematic step (before introducing constitutive assumptions) because the strain rate constructed from inner-product geometry is symmetric and has no antisymmetric part. We further show that when the fluid arises as a thin-shell limit of an ambient three-dimensional flow, the operator that emerges depends on the boundary condition imposed in the normal direction: stress-free (Navier slip) conditions recover the deformation Laplacian, while Hodge boundary conditions recover the Hodge Laplacian, via an explicit decomposition of the ambient Bochner Laplacian into intrinsic and extrinsic pieces. The intrinsic piece is the deformation Laplacian regardless of the boundary condition. As an analytical confirmation, we show that the kinematic selection is consistent with the known failure of the energy inequality for the Hodge Laplacian on the hyperbolic plane $\HH^2$: the deformation Laplacian is coercive on $\HH^2$ while the Hodge Laplacian is not, because the Ricci term has the opposite sign in the two operators. We further prove 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, resolving the analytical obstruction preventing the corresponding result for the Hodge Laplacian.