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

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 $p\in (1,\mathrm{\infty })$ 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.