Existence and equilibration of global weak solutions to kinetic models for dilute polymers II: Hookean-type models

摘要

We show the existence of global-in-time weak solutions to a general class of coupled Hookean-type bead-spring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible NavierStokes equations in a bounded domain in ~d, d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a FokkerPlanck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term. We require no structural assumptions on the drag term in the FokkerPlanck equation; in particular, the drag term need not be corotational. With a square-integrable and divergence-free initial velocity datum _0 for the NavierStokes equation and a non-negative initial probability density function ψ _0 for the FokkerPlanck equation, which has finite relative entropy with respect to the Maxwellian M, we prove, via a limiting procedure on certain regularization parameters, the existence of a global-in-time weak solution t → ((t), ψ(t)) to the coupled NavierStokesFokkerPlanck system, satisfying the initial condition ((0), ψ(0)) = (_0, ψ _0), such that t → (t) belongs to the classical Leray space and t → ψ(t) has bounded relative entropy with respect to M and t → ψ(t)/M has integrable Fisher information (with respect to the measure ν:= M(q)dqdx) over any time interval [0, T], T>0. If the density of body forces f on the right-hand side of the NavierStokes momentum equation vanishes, then a weak solution constructed as above is such that t → ((t), ψ(t)) decays exponentially in time to 0, M) in the L ~2 × L ~1-norm, at a rate that is independent of (_0, ψ _0) and of the center-of-mass diffusion coefficient. Our arguments rely on new compact embedding theorems in Maxwellian-weighted Sobolev spaces and a new extension of the KolmogorovRiesz theorem to Banach-space-valued Sobolev spaces.