# Existence of global weak solutions to the kinetic Hookean dumbbell model for incompressible dilute polymeric fluids

## TL;DR

This paper proves the existence of global weak solutions for the kinetic Hookean dumbbell model in 2D and weak subsolutions in 3D, linking microscopic polymer models to macroscopic fluid equations.

## Contribution

It establishes the existence of large-data global weak solutions for the Hookean dumbbell model in two dimensions and introduces the concept of weak subsolutions with defect measures in three dimensions.

## Key findings

- Existence of large-data global weak solutions in 2D.
- Existence of weak subsolutions with defect measures in 3D.
- Rigorous connection between the Hookean dumbbell model and the Oldroyd-B model.

## Abstract

We explore the existence of global weak solutions to the Hookean dumbbell model, a system of nonlinear partial differential equations that arises from the kinetic theory of dilute polymers, involving the unsteady incompressible Navier--Stokes equations in a bounded domain in two or three space dimensions, coupled to a Fokker--Planck-type parabolic equation. We prove the existence of large-data global weak solutions in the case of two space dimensions. Indirectly, our proof also rigorously demonstrates that, in two space dimensions at least, the Oldroyd-B model is the macroscopic closure of the Hookean dumbbell model. In three space dimensions, we prove the existence of large-data global weak subsolutions to the model, which are weak solutions with a defect measure, where the defect measure appearing in the Navier--Stokes momentum equation is the divergence of a symmetric positive semidefinite matrix-valued Radon measure.