Existence and uniqueness results for the Doi–Edwards polymer melt model: the case of the (full) nonlinear configurational probability density equation

Abstract

This paper proves the existence and uniqueness of non-negative solutions of the (full) nonlinear configurational probability diffusion equation-which is of Fokker-Planck-Smoluchowski type-of the Doi-Edwards model for polymer melt rheology. This is achieved using the Schauder fixed point theorem and Galerkin's approximation method.

Solvability for a drift-diffusion system with Robin boundary conditions

“…When coupled with fluid flows equations, the resulting systems are usually quite complex due to the micro-macro effect. A short reference list is [8,9,10,11,15,22] for some problems arising in different contex, including the theory of dilute or melt polymers. Apart from the theory of stochastic process -mainly the Fokker-Planck equation -a priviledged field of application is the theory of semi conductors.…”

Existence and uniqueness of a density probability solution for the stationary Doi–Edwards equation

“…F (s = 0) = F (s = 1) = (1/4π) (1.2) and for the initial condition: F (t = 0) = F 0 (s, u) (1.3) (see [9], [17] and [5]). In the equation (1.1) D > 0 and ǫ ≥ 0 are physical coefficients and κ = κ(t) ∈ M 3 (R) is the velocity gradient; we also have…”

“…In order to prove this last result, we establish that solutions of problem (1.4) − (1.5) are the limits when t → ∞ of solutions of the time dependent Doi Edwards problem. Since the solutions of the Doi Edwards problem are known to be probability densities (see [5]), this provides the result; this approach also provides the desired uniqueness (see Section 5 and Section 6). The main difficulty in the proof is to bound on R + t in a suitable norm nonlinear terms such as ∂ ∂s F κ : λ(F ) .…”

Mathematical Existence Results for the Doi–Edwards Polymer Model

“…Nonetheless we can cite some recent theoretical papers on this subject, see [4,19], in which the authors are only interested in specific cases: one dimensional shear flows under the independent alignment assumption in [19], flows for which the coupling between the velocity and the stress is not taking into account, see [4]. More generally, there seems to be a real challenge to obtain global existence in time for models of polymers.…”