Weighted $\boldsymbol{H}^2$ regularity of fluid-structure interaction when the interface intersects the boundary

This paper analyzes the regularity of solutions to a fluid-structure interaction (FSI) problem involving a Stokesian fluid and a linear elastic solid in a two-dimensional polygonal domain, seperated by an interface which intersects the boundary of the domain. The main challenges arise from the limited solution regularity caused by geometric singularities at the domain corners. To address this, we introduce tailored weighted Sobolev spaces, denoted by $\mathbf{H}^{k,l}_{\boldsymbol{\beta}}$, and a novel solution operator framework on these spaces. This framework provides the key insight for decoupling the coupled FSI system into tractable fluid and elastic solid subproblems in the regularity analysis. As our main result, we prove the existence and uniqueness of a solution in the weighted Sobolev space $\mathbf{H}^{2,2}_{\boldsymbol{\beta}}$.