$\mathrm{L}^p$-based Sobolev theory for PDEs on closed manifolds of class $C^m$

We study the $\mathrm{L}^p$-based ($1<p<\infty$) well-posedness and Sobolev regularity for appropriate weak formulations of the (stationary) Laplace--Beltrami, Bochner Laplace, tangent Stokes, and tangent Navier--Stokes problems on compact, connected $d$-dimensional manifolds without boundary of class $C^m$ ($m \geq 2$) embedded in $\mathrm{R}^{d+1}$, in terms of the regularity of the source terms and the manifold. Firstly, we localize and translate the Laplace--Beltrami problem into flat domains to establish $\mathrm{L}^p$-based ($2 \leq p < \infty$) Sobolev regularity results as a consequence of the powerful Calderón-Zygmund theory. From then onwards, our strategy is purely ``variational'', building upon the abstract functional analytic idea of ``duality'' and classical results such as the Banach--Nečas--Babuška theorem and the generalized Babuška--Brezzi theory in reflexive Banach spaces to analyze and relate to each other different strong, weak and ultra-weak formulations for the aforementioned problems. In particular, by exploiting the lack of manifold boundary, we are able to decouple the velocity and the pressure unknowns in the tangent Stokes problem to establish their higher-regularity $\mathbf{W}^{k,p} \times \mathrm{W}^{k-1,p}$ ($k \geq 2$) as a consequence of the $\mathrm{L}^p$-based well-posedness and regularity theory for the Laplace--Beltrami and Bochner--Laplace operators. We finish this work by applying this newly derived regularity theory to: i) define and appropriate Stokes operator and study its spectral and regularity properties, ii) establish $\mathrm{L}^p$-based ($1<p<\infty$) existence and well-posedness for the (stationary) tangent Oseen and Navier--Stokes equations via a Galerkin method and bootstrapping, and higher $\mathrm{L}^p$-based Sobolev regularity results. iii) transfer our regularity results to other choices of vector Laplace operators.