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

Gonzalo A. Benavides, Ricardo H. Nochetto, Mansur Shakipov

Abstract

This paper is the second part of a two-paper series, initiated in Benavides, Nochetto and Shakipov arXiv:XXXX.XXXXX 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 Benavides, Nochetto and Shakipov arXiv:XXXX.XXXXX, 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 $\mathbf{W}^{m,p} \times \mathrm{W}^{m-1,p}$ ($m \geq 2$) 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.

$\mathrm{L}^p$-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 Benavides, Nochetto and Shakipov arXiv:XXXX.XXXXX for scalar PDEs on hypersurfaces, and is concerned with the well-posedness and -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 -dimensional manifolds without boundary embedded in . We prove -regularity for any for manifolds of minimal regularity or for . Building upon the -based scalar elliptic theory from Benavides, Nochetto and Shakipov arXiv:XXXX.XXXXX, 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 () as a consequence of the -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 and . We next extend the well-posedness to and prove higher-order -based regularity. We finally examine alternative choices to the Bochner Laplace operator that are useful in fluid dynamics.

Paper Structure

This paper contains 22 sections, 47 theorems, 241 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Statements of the main results
  4. Preliminaries
  5. Differential geometry and Sobolev spaces on Γ
  6. The Laplace-Beltrami problem
  7. The Bochner--Laplace problem
  8. Lᵖ-based Sobolev regularity
  9. Tangent Stokes equations
  10. Helmholtz--Weyl decomposition of Wᵐᵖ
  11. Well-posedness in W¹ᵖ x Lᵖ
  12. Higher Lᵖ-based Sobolev regularity
  13. The Stokes operator
  14. Tangent Navier--Stokes equations
  15. Tangent Oseen equations
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\Gamma$ be of class $C^2$ and let $p \in (1,\infty)$. Then, for every $\bf \in (\mathbf{W}^{1,p^*}_t(\Gamma))'$, there exists a unique $\mathbf{u} \in \mathbf{W}^{1,p}_t(\Gamma)$ such that Moreover, there exists a positive constant $C$, depending only on $\Gamma$ and $p$, such that Furthermore, if $\Gamma$ is of class $C^{m+2,1}$ for some nonnegative integer $m$ and $\bf \in \mathbf{W}^{m,p

Theorems & Definitions (82)

  • Theorem 1.1: Bochner--Laplace
  • Theorem 1.2: tangent Stokes
  • Theorem 1.3: existence of solutions of the incompressible tangent Navier--Stokes equations
  • Theorem 1.4: tangent Navier--Stokes with small data
  • Theorem 1.5: higher regularity of the tangent Navier--Stokes equations
  • Definition 2.1: Sobolev spaces on $\Gamma$
  • Lemma 2.2: product rule on $\mathrm{W}^{m,p}(\Gamma)$
  • Theorem 2.3: density
  • Proposition 2.4: equivalent norm in $\mathrm{W}^{m,p}(\Gamma)$
  • Remark 2.5: $\mathrm{W}^{m,p}(\Gamma)$ coincides with $H^p_m(\Gamma)$ of Hebey and Robert HebeyRobert2008
  • ...and 72 more