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

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.