Existence of a local strong solution to the beam–polymeric fluid interaction system

Abstract

We construct a unique local strong solution to the finitely extensible nonlinear elastic (FENE) dumbbell model of Warner-type for an incompressible polymer fluid (described by the Navier–Stokes–Fokker–Planck equations) interacting with a flexible elastic shell. The latter occupies the flexible boundary of the polymer fluid domain and is modeled by a beam equation coupled through kinematic boundary conditions and the balance of forces. In the 2D case for the co-rotational Fokker–Planck model we obtain global-in-time strong solutions.

A main step in our approach is the proof of local well-posedness for just the solvent–structure system in higher-order topologies which is of independent interest. Different from most of the previous results in the literature, the reference spatial domain is an arbitrary smooth subset of $R^3$, rather than a flat one. That is, we cover viscoelastic shells rather than elastic plates. Our results also supplement the existing literature on the Navier–Stokes–Fokker–Planck equations posed on a fixed bounded domain.