Existence of global weak solutions to compressible isentropic finitely extensible nonlinear bead-spring chain models for dilute polymers: The two-dimensional case

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:261

Article

Barrett, John W.¹  Suli, Endre² 

[1] Univ London Imperial Coll Sci Technol & Med, Dept Math, Huxley Bldg, London SW7 2AZ, England

[2] Univ Oxforr4, Math Inst, Oxford OX2 6GG, England

关键词: Kinetic polymer models;    FENE chain;    Compressible Navier-Stokes-Fokker-Planck system;    Variable density;    Nonhomogeneous dilute polymer;   

DOI  :  10.1016/j.jde.2016.03.018

来源: Elsevier

【 摘 要 】

We prove the existence of global-in-time weak solutions to a general class of models that arise from 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 class of models under consideration involves the unsteady, compressible, isentropic, isothermal Navier-Stokes system in a bounded domain Omega in R-d, d = 2, for the density rho, the velocity (u) under tilde and the pressure p of the fluid, with an equation of state of the form p(rho) = c(p)rho(gamma), where c(p) is a positive constant and gamma > 1. The right-hand side of the Navier- Stokes momentum equation includes an elastic extra-stress tensor, which is the classical Kramers expression. The elastic extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker-Planck-type parabolic equation, a crucial feature of which is the presence of a centre-of-mass diffusion term. This extends the result in our paper J.W. Barrett and E. Still (2016) [9], which established the existence of global-in-time weak solutions to the system for d is an element of {2, 3} and gamma > 3/2, but the elastic extra-stress tensor required there the addition of a quadratic interaction term to the classical Kramers expression to complete the compactness argument on which the proof was based. We show here that in the case of d = 2 and gamma > 1 the existence of global-in-time weak solutions can be proved in the absence of the quadratic interaction term. Our results require no structural assumptions on the drag term in the Fokker Planck equation; in particular, the drag term need not be corotational. With a nonnegative initial density rho(0) is an element of L-infinity (Omega)for the continuity equation; a square-integrable initial velocity datum (u) under tilde (0) for the Navier Stokes momentum equation; and a nonnegative initial probability density function psi 0 for the Fokker-Planck equation, which has finite relative entropy with respect to the Maxwellian M associated with the spring potential in the model, we prove, via a limiting procedure on a pressure regularization parameter, the existence of a global-in-time bounded-energy weak solution t bar right arrow (rho(t), (u) under tilde (t), psi(t)) to the coupled Navier-Stokes-Fokker-Planck system, satisfying the initial condition (rho(0), (u) under tilde (0), psi(0)) = (rho(0), (u) under tilde (0), psi(0)). (C) 2016 Elsevier Inc. All rights reserved.

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