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

Abstract

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} $\Omega$ using \emph{the method of layer potentials} on $M \supset \Omega$. The operator $\bsXi_{0, 0}$ is the classical Stokes operator. Under suitable positivity assumptions on $V$ and $V_{0}$, we prove that $\bsXi$ is Fredholm. This allows us then to define the single- and double-layer potentials $\bsS$ and $\frac12 + \bsK$. Under further positivity assumptions, we prove that $\bsS$ and $\frac12 + \bsK$ are also Fredholm. Under slightly stronger assumptions on $V$ and $V_{0}$, we prove \emph{the invertibility} of the operators $\bsXi$, $\bsS$, and $\frac12 + \bsK$. The invertibility of these operators leads to \emph{well-posedness results} for the associated (linear) Stokes boundary value problem with Dirichlet boundary conditions on $\Omega$. 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.