Existence of Large-Data Global Weak Solutions to Kinetic Models of Nonhomogeneous Dilute Polymeric Fluids

TL;DR

This paper establishes the existence of global weak solutions for complex kinetic models of nonhomogeneous dilute polymeric fluids, combining advanced mathematical techniques to handle the coupled Navier-Stokes and Fokker-Planck equations.

Contribution

It proves the existence of large-data global weak solutions for a broad class of coupled bead-spring chain models with nonlinear elastic springs in nonhomogeneous fluids.

Findings

Existence of global weak solutions for nonhomogeneous polymeric fluid models.

Application of Galerkin approximation and compactness methods.

Handling of density-dependent coefficients in the coupled system.

Abstract

We prove the existence of large-data global-in-time weak solutions to a general class of coupled bead-spring chain models with finitely extensible nonlinear elastic (FENE) type spring potentials for nonhomogeneous incompressible dilute polymeric fluids in a bounded domain in $\mathbb{R}^d$, $d=2$ or $3$. The class of models under consideration involves the Navier--Stokes system with variable density, where the viscosity coefficient depends on both the density and the polymer number density, coupled to a Fokker--Planck equation with a density-dependent drag coefficient. The proof is based on combining a truncation of the probability density function with a two-stage Galerkin approximation and weak compactness and compensated compactness techniques to pass to the limits in the sequence of Galerkin approximations and in the truncation level.