On the well-posedness of a nonlocal kinetic model for dilute polymers with anomalous diffusion

Abstract In this work, we study a class of nonlocal-in-time kinetic models of incompressible dilute polymeric fluids. The system couples a macroscopic balance of linear momentum equation with a mesoscopic subdiffusive Fokker–Planck equation governing the evolution of the probability density function (PDF) of polymers. The model incorporates nonlocal features to capture subdiffusive and memory-type phenomena. Our main result asserts the existence of global-in-time large-data weak solutions to this nonlocal system. The proof relies on an energy estimate involving a suitable relative entropy, which enables us to handle the critical general non-corotational drag term that couples the two equations. Crucial steps in our analysis are the proof of the nonnegativity of the PDF and establishing strong convergence of the sequence of Galerkin approximations. This involves a novel compactness result for nonlocal PDEs. Lastly, we prove the uniqueness of weak solutions with sufficient regularity.