Dissipative weak solutions to compressible Navier-Stokes-Fokker-Planck systems with variable viscosity coefficients

TL;DR

This paper extends the existence theory of weak solutions for a coupled compressible Navier-Stokes and Fokker-Planck system to include variable viscosity coefficients depending on polymer density, demonstrating weak sequential stability.

Contribution

It introduces a framework for proving the weak sequential stability of dissipative solutions with density-dependent viscosities in polymeric fluid models.

Findings

Established weak sequential stability of solutions with variable viscosities.

Extended previous constant-viscosity results to density-dependent cases.

Provided conditions linking viscosity behavior and pressure at high densities.

Abstract

Motivated by a recent paper by Barrett and S\"uli [J.W. Barrett & E. S\"uli: Existence of global weak solutions to compressible isentropic finitely extensible bead-spring chain models for dilute polymers, Math. Models Methods Appl. Sci., 26 (2016)], we consider the compressible Navier--Stokes system coupled with a Fokker--Planck type equation describing the motion of polymer molecules in a viscous compressible fluid occupying a bounded spatial domain, with polymer-number-density-dependent viscosity coefficients. The model arises in the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids, where the polymer molecules are idealized as bead-spring chains with finitely extensible nonlinear elastic (FENE) type spring potentials. The motion of the solvent is governed by the unsteady, compressible, barotropic Navier--Stokes system, where the viscosity coefficients in the Newtonian stress tensor depend on the polymer number density. Our goal is to show that the existence theory developed in the case of constant viscosity coefficients can be extended to the case of polymer-number-density-dependent viscosities, provided that certain technical restrictions are imposed, relating the behavior of the viscosity coefficients and the pressure for large values of the solvent density. As a first step in this direction, we prove here the weak sequential stability of the family of dissipative (finite-energy) weak solutions to the system.