Global Existence Results for Some Viscoelastic Models with an Integral Constitutive Law

Abstract

We provide a proof of global regularity of solutions of some models of viscoelastic flow with an integral constitutive law, in the two spatial dimensions and in a periodic domain. Models that are included in these results are classical models for flow memory: for instance some K-BKZ models, the PSM model or the Wagner model. The proof is based on the fact that these models naturally give a L ∞ -bound on the stress and that they allow to control the spatial gradient of the stress. The main result does not cover the case of the Oldroyd-B model.

Mathematical Existence Results for the Doi–Edwards Polymer Model

“…Thus, for the FENE type models, N. Masmoudi [27] proved a global existence result in dimension 2. Similarly, for integral fluid of type K-BKZ such results hold too (see [3]). …”

“…More precisely we prove the following Proposition (the proof is detailed in [3] and [6]). 2 then for all t P p0, t ‹ q we have…”

“…For such a model the global existence result is a consequence of a general result on viscoelastic flows with memory, see [3]. Nevertheless, it is also well known that this approximation causes serious error in certain situations, this is clearly specified in the seminal book [14].…”

“…(5.13) 3 The term "standard" refers to models like K-BKZ in which the memory is a given function, usually on exponential type: KpT q " e´T .…”

Equations for Polymeric Materials

“…; t; x/ contains all the information about the past deformation of the polymers in the fluid and hence depends on derivatives of the velocity. Of course one should impose some assumption of the functional F. Some of the well-known models in the literature are the Wagner model, the PSM model, and the K-BKZ models (see [15] for the mathematical analysis of such models).…”