$\mathrm{L}^p$-based Sobolev theory on closed manifolds of minimal regularity: Vector-valued problems

TL;DR

This paper develops an $ ext{L}^p$-based Sobolev theory for vector-valued PDEs on manifolds with minimal regularity, including fluid dynamics equations, providing new regularity and well-posedness results.

Contribution

It introduces a variational, parametrization-free approach to establish $ ext{W}^{m,p}$-regularity for vector PDEs on low-regularity manifolds, extending scalar theories to vector-valued problems.

Findings

Proved $ ext{W}^{m,p}$-regularity for manifolds of minimal regularity.

Established higher-regularity solutions for Stokes and Navier--Stokes equations.

Analyzed spectral properties of the Stokes operator and solution existence for Navier--Stokes.

Abstract

This paper is the second part of a two-paper series, initiated in arXiv:2603.02163 for scalar PDEs on hypersurfaces, and is concerned with the well-posedness and $\mathrm{L}^p$-based Sobolev regularity of vector-valued PDEs of interest in fluid dynamics. This family of PDEs includes the (stationary) Bochner Laplace, tangent Stokes and Oseen, and tangent Navier--Stokes equations. We present several strong, weak and ultra-weak formulations of these problems on compact, connected $d$-dimensional manifolds without boundary embedded in $\mathrm{R}^{d+1}$. We prove $\mathrm{W}^{m,p}$-regularity for any $p \in (1,\infty)$ for manifolds of minimal regularity $C^{m+1}$ or $C^{m,1}$ for $m\ge1$. Building upon the $\mathrm{L}^p$-based scalar elliptic theory from arXiv:2603.02163, we develop a parametrization-free and purely variational approach that resorts to classical results such as the Banach--Ne\v{c}as--Babu\v{s}ka theorem and the generalized Babu\v{s}ka--Brezzi theory in reflexive Banach spaces. In particular, by exploiting the manifold closedness, we decouple the velocity and pressure variables in the tangent Stokes problem to establish their higher-regularity $\mathbf{W}^{m,p} \times \mathrm{W}^{m-1,p}$ ($m \geq 2$) as a consequence of the $\mathrm{L}^p$-based well-posedness and regularity theory for the Laplace--Beltrami and Bochner--Laplace operators. We study spectral and regularity properties of an appropriate Stokes operator, and apply them to show existence of solutions for the Navier--Stokes equations for $p=2$ and $d \leq 4$. We next extend the well-posedness to $p > 2$ and prove higher-order $\mathrm{L}^p$-based regularity. We finally examine alternative choices to the Bochner Laplace operator that are useful in fluid dynamics.