Newtonian limit for weakly viscoelastic fluid flows

This article addresses the low Weissenberg asymptotic analysis (Newtonian limit) of some macroscopic models of viscoelastic fluid flows in the framework of global weak solutions. We investigate the convergence of the corotational Johnson-Segalman, the FENE-P (closure proposed by Peterlin of the Finitely Extensible Nonlinear Elastic model), and the Giesekus and Phan-Thien and Tannes models. Relying on a priori bounds coming from energy or free energy estimates, we first study the weak convergence toward the Navier-Stokes system. We then turn to the main focus of our paper, i.e., the strong convergence. The novelty of our work is to address these issues by relative entropy estimates, which require the introduction of some corrector terms. We also take into account the presence of defect measures in the initial data, uniform with respect to the Weissenberg number, and prove that they do not perturb the Newtonian limit of the corotational system.