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

TL;DR

This paper establishes the existence, nonnegativity, and uniqueness of weak solutions for a nonlocal-in-time kinetic model describing dilute polymeric fluids with subdiffusive behavior, using energy estimates and compactness techniques.

Contribution

It introduces a novel analysis framework for nonlocal kinetic models with memory effects, proving global existence and uniqueness of solutions.

Findings

Existence of global-in-time weak solutions

Nonnegativity of the probability density function

Uniqueness of solutions with sufficient regularity

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 mezoscopic subdiffusive Fokker-Planck equation governing the evolution of the probability density function of polymer configurations. 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. As a side result, we prove nonnegativity of the probability density function. A crucial step in our analysis is to establish strong convergence of the sequence of Galerkin approximations by a combination of techniques, involving a novel compactness result for nonlocal PDEs. Lastly, we prove the uniqueness of weak solutions with sufficient regularity.