Analysis of Navier--Stokes--Fokker--Planck systems for incompressible dilute polymeric fluids - He, C (2023)
We show the existence of large-data global-in-time weak solutions to various classes of coupled bead-spring chain models with finitely extensible nonlinear elastic (FENE) type spring potentials for incompressible dilute polymeric fluids in a bounded domain in Rd, d = 2 or 3. The governing system consists of the transport equation and the Navier--Stokes equation coupled to the Fokker--Planck equation through the elastic extra-stress tensor which is defined by the Kramers expression. The proofs are based on truncating the probability density function and approximating with Galerkin semi-discretizations on the spatial domains. We derive uniform bounds independent of the Galerkin and truncation parameters; then with weak compactness and compensated compactness techniques, we pass to the limits in the sequences of Galerkin approximations and in the truncation level. The technical tools involve using Nikolskiĭ norm estimates to derive uniform estimates for fractional time derivatives and using various generalizations of the Aubin--Lions Lemma to deduce strong convergence of approximating sequences. We also apply the Div-Curl Lemma and Vitali's Convergence Theorem to deduce the strong convergence of the sequence of approximations of the probability density function in $L^1$.