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

Dominic Breit and Prince Romeo Mensah

Mathematische Nachrichten, 2025, vol. 298, issue 7, 2327-2379

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 R3$\mathbb {R}^3$, 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.

Date: 2025
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