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Well-posedness of a generalized Stokes operator on smooth bounded domains via layer-potentials

Mirela Kohr, Victor Nistor, Wolfgang L. Wendland

TL;DR

The paper develops a layer-potential framework for a generalized Stokes operator ${\boldsymbol \Xi}_{V,V_{0}}$ on smooth manifolds and proves its Fredholmness and invertibility properties under natural positivity and geometric assumptions. Central to the approach is the Moore–Penrose pseudoinverse ${\boldsymbol \Xi}^{(-1)}$, which yields explicit single and double layer potentials ${\mathcal S}_{\rm ST}$ and ${\mathcal D}_{\rm ST}$ with well-defined boundary traces and jump relations. The authors establish ellipticity and self-adjointness of the boundary operators ${\boldsymbol S}$ and ${\tfrac{1}{2}+{\boldsymbol K}}$, and, under suitable non-vanishing conditions on the potentials, obtain invertibility on appropriate subspaces, enabling solvability of Dirichlet problems for the generalized Stokes system. The results extend classical Stokes theory to a broad manifold setting, providing precise mapping properties, boundary behavior, and a robust boundary-integral formulation for well-posed boundary-value problems.

Abstract

We prove the invertibility of the relevant single and double layer potentials associated to some generalizations of the Stokes operator on bounded domains. In order to do that, we first develop an ``algebra tool kit'' to deal with limit and jump relations of layer operators. We do that first on $\mathbb{R}^{n}$ for operators acting on a distribution supported on $\{x_{n} = 0\}$ and then in general on (possibly non-compact manifolds). We use these results to study the limit and jump relations of the layer potential operators associated to our generalized Stokes operators. In turn, we then use these results to prove the Fredholm property of single and double layer potentials of the generalized Stokes operator and even their invertibility when the auxiliary potentials satisfy suitable non-vanishing conditions. As an application, we obtain well-posedness results.

Well-posedness of a generalized Stokes operator on smooth bounded domains via layer-potentials

TL;DR

The paper develops a layer-potential framework for a generalized Stokes operator on smooth manifolds and proves its Fredholmness and invertibility properties under natural positivity and geometric assumptions. Central to the approach is the Moore–Penrose pseudoinverse , which yields explicit single and double layer potentials and with well-defined boundary traces and jump relations. The authors establish ellipticity and self-adjointness of the boundary operators and , and, under suitable non-vanishing conditions on the potentials, obtain invertibility on appropriate subspaces, enabling solvability of Dirichlet problems for the generalized Stokes system. The results extend classical Stokes theory to a broad manifold setting, providing precise mapping properties, boundary behavior, and a robust boundary-integral formulation for well-posed boundary-value problems.

Abstract

We prove the invertibility of the relevant single and double layer potentials associated to some generalizations of the Stokes operator on bounded domains. In order to do that, we first develop an ``algebra tool kit'' to deal with limit and jump relations of layer operators. We do that first on for operators acting on a distribution supported on and then in general on (possibly non-compact manifolds). We use these results to study the limit and jump relations of the layer potential operators associated to our generalized Stokes operators. In turn, we then use these results to prove the Fredholm property of single and double layer potentials of the generalized Stokes operator and even their invertibility when the auxiliary potentials satisfy suitable non-vanishing conditions. As an application, we obtain well-posedness results.

Paper Structure

This paper contains 27 sections, 39 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Normal lateral limits at $x_{n} = 0$ of pseudodifferential operators on $\mathbb R^{n}$
  3. Motation: traces, normal lateral limits, and more
  4. Lateral limits of pseudodifferential operators of arbitrary orders
  5. Lateral limits of pseudodifferential operators of orders $<-1$
  6. Lateral limits of pseudodifferential operators of orders $-1$
  7. Normal lateral limits and abstract jump relations on manifolds
  8. Normal tubular neighborhoods
  9. Lateral limits on manifolds for operators of order $m < -1$
  10. Lateral limits on manifolds for operators of order $m = -1$
  11. Mapping properties
  12. The deformation and Stokes operators and Green formulas
  13. A general integration by parts formula
  14. Diferential operators
  15. Green formulas on $\Omega$
  16. ...and 12 more sections

Key Result

Lemma 2.1

Let $h$ be a locally integrable function on $\Gamma := \{x_{n} = 0\} \subset \mathbb R^{n}$. Then we define the distribution $h \delta_{\Gamma} := \boldsymbol h \delta_{\Gamma}$ on $\mathbb R^{n}$ by the formula If $h \in L^{2}({\Gamma})$, then $h \delta_{\Gamma} \in H^{s'}(\mathbb R^{n})$ for all $s' < -\frac{1}{2}$. This definition extends by duality to $h \in H^{s}(\Gamma; E)$, in which case $

Theorems & Definitions (77)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.6
  • Remark 2.7
  • Definition 2.8
  • Remark 2.9
  • Proposition 2.10
  • ...and 67 more