Well-posedness of a generalized Stokes operator on domains with cylindrical ends via layer-potentials

We study the \emph{generalized Stokes operator} \begin{equation*} \bsXi \ede \bsXi _{V,V_0} \ede \left(\begin{array}{ccc} \bsL + V & \nabla \ \nabla^* & -V_0 \end{array}\right) \end{equation*} on a \emph{domain with straight cylindrical ends} $\mathrm{\Omega }$ using \emph{the method of layer potentials} on $M\supset \mathrm{\Omega }$. The operator ${\text{bsXi}}_{0,0}$ is the classical Stokes operator. Under suitable positivity assumptions on $V$ and $V_0$, we prove that $\text{bsXi}$ is Fredholm. This allows us then to define the single- and double-layer potentials $\text{bsS}$ and $\frac{1}{2}+\text{bsK}$. Under further positivity assumptions, we prove that $\text{bsS}$ and $\frac{1}{2}+\text{bsK}$ are also Fredholm. Under slightly stronger assumptions on $V$ and $V_0$, we prove \emph{the invertibility} of the operators $\text{bsXi}$, $\text{bsS}$, and $\frac{1}{2}+\text{bsK}$. The invertibility of these operators leads to \emph{well-posedness results} for the associated (linear) Stokes boundary value problem with Dirichlet boundary conditions on $\mathrm{\Omega }$. The proofs of these results required us to develop many related tools. In particular, we develop an algebra tool kit'' to deal with \emph{limit and jump relations of layer potentials.} We also develop Green formulas and energy estimates for our generalized Stokes operator $\bsXi$ on manifolds with straight cylindrical ends, which requires a careful geometric study of the related differential operators, such as the deformation operator $\Def$. For completeness, we review suitable classes of pseudodifferential operators on manifolds with straight cylindrical ends that were studied in some previous papers of ours (including The Stokes operator on manifolds with cylindrical ends,‘’ J. Diff. Equations, 2024). As an application, we prove the well-posedness result for the Dirichlet problem for the generalized Navier-Stokes system with small data on a domain with cylindrical ends.