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

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