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

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 $m\ge 1$. 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--Nečas--Babuška theorem and the generalized Babuška--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 ${\mathbf{W}}^{m,p}\times {\mathrm{W}}^{m-1,p}$ ($m\ge 2$) 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 $d\le 4$. 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.

v2: We streamline the overall arguments of the paper and improve its results. Given its length, we splitted the manuscript in two parts: arXiv:2603.02163 and the present manuscript. 43 pages v3: Metadata update. No changes in the content of the manuscript