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

TL;DR

This paper establishes the invertibility and well-posedness of generalized Stokes operators on smooth bounded domains by analyzing layer potentials and their limit and jump relations, extending classical results to more general settings.

Contribution

It develops an algebraic toolkit for layer potential analysis and proves invertibility of layer operators for generalized Stokes problems on complex domains.

Findings

Proves invertibility of layer potentials for generalized Stokes operators.

Establishes the Fredholm property and invertibility under certain conditions.

Provides well-posedness results for the generalized Stokes problem.

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.