Fluid Problems Posed on Closed Hypersurfaces: Circumventing the Saddle-Point Structure

Abstract

For a d-dimensional closed hypersurface in ℝ^d+1, we reformulate the (stationary) tangent Stokes, transient tangent Stokes, and transient tangent Navier-Stokes equations by replacing the divergence equation with a pressure Poisson equation, thereby circumventing their saddle-point structure. The pressure Poisson equation hinges on a commutator identity between the divergence and the surface diffusion operator, whose lower-order term depends on the Ricci curvature, and requires no boundary conditions for the pressure. We show that the reformulated operator is the composition of a linear, block-triangular data map with the primal operator. Consequently, each reformulation is well-posed if and only if the primal problem is, and solving it with arbitrary data amounts to solving the primal problem with a modified divergence datum. For hypersurfaces of class C² and data of the pressure Poisson equation in H⁻¹, the tangent Stokes reformulations can be posed in H¹-based spaces, which paves the way for conforming discretizations that do not require a discrete inf-sup condition.

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BibTeX RIS

BibTeX

@misc{nochetto2026fluid,
  title = {{Fluid Problems Posed on Closed Hypersurfaces: Circumventing the Saddle-Point Structure}},
  author = {Ricardo H. Nochetto and Mansur Shakipov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.36641},
  eprint = {2610.36641},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.36641},
}

RIS

TY  - UNPB
TI  - Fluid Problems Posed on Closed Hypersurfaces: Circumventing the Saddle-Point Structure
AU  - Ricardo H. Nochetto
AU  - Mansur Shakipov
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.36641
UR  - https://arxiv.org/abs/2610.36641
AB  - For a d-dimensional closed hypersurface in ℝ^d+1, we reformulate the (stationary) tangent Stokes, transient tangent Stokes, and transient tangent Navier-Stokes equations by replacing the divergence equation with a pressure Poisson equation, thereby circumventing their saddle-point structure. The pressure Poisson equation hinges on a commutator identity between the divergence and the surface diffusion operator, whose lower-order term depends on the Ricci curvature, and requires no boundary conditions for the pressure. We show that the reformulated operator is the composition of a linear, block-triangular data map with the primal operator. Consequently, each reformulation is well-posed if and only if the primal problem is, and solving it with arbitrary data amounts to solving the primal problem with a modified divergence datum. For hypersurfaces of class C^2 and data of the pressure Poisson equation in H^−1, the tangent Stokes reformulations can be posed in H^1-based spaces, which paves the way for conforming discretizations that do not require a discrete inf-sup condition.
ER  -

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