Analysis of Navier--Stokes--Fokker--Planck systems for incompressible dilute polymeric fluids

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 Nikolski\u{\i} 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 $L^1$. We first focus on homogeneous dilute polymeric fluids. The key feature is the polymer-number-density-dependent viscosity coefficient appearing in the Navier--Stokes equation. Then we move on to nonhomogeneous dilute polymeric fluids, featuring the presence of a density-dependent and polymer-number-density-dependent viscosity coefficient in the Navier--Stokes equation and a density-dependent drag coefficient in the Fokker--Planck equation. Finally, we consider nonisothermal homogeneous dilute polymeric fluids. To complete the nonisothermal Navier--Stokes--Fokker--Planck system, the temperature evolution equation is also introduced to form a thermodynamically consistent model. To simplify the nonisothermal model, we present the existence proof ...