Global existence results for some viscoelastic models with an integral constitutive law (Q2878984)

This paper is concerned with the global existence of solutions to some models of viscoelastic fluids with an integral constitutive law in a two-dimensional periodic domain. The models are the incompressible Navier-Stokes equations coupled with a transport equation for the deformation tensor through the stress-strain relation, and include classical models for flow memory: for instance, some K-BKZ models, the PSM model, or the Wagner model. Under certain assumptions on the integral kernel in the stress tensor, the author proves the global existence and uniqueness of spatially periodic (strong) solutions. The proof is based on the fact that these models naturally give an \(L^\infty\)-bound on the stress and that they allow one to control the spatial gradient of the stress. It should be remarked that the main result in this paper does not cover the case of the Oldroyd-B model.