Global existence and optimal decay rate of weak solutions to the co-rotation Hooke dumbbell model

TL;DR

This paper proves the global existence and optimal decay rates of weak solutions for a coupled Navier-Stokes and Fokker-Planck model describing polymer fluids, using defect measure and Fourier splitting methods.

Contribution

It establishes the global existence and decay rates for weak solutions to the co-rotation Hooke dumbbell model, a micro-macro fluid dynamics system.

Findings

Global weak solutions exist under various initial conditions.

Optimal L^2 decay rates are achieved for these solutions.

The methods used include defect measure propagation and Fourier splitting.

Abstract

In this paper, we mainly study global existence and optimal $L^2$ decay rate of weak solutions to the co-rotation Hooke dumbbell model. This micro-macro model is a coupling of the Navier-Stokes equation with a nonlinear Fokker-Planck equation. Based on the defect measure propagation method, we prove that the co-rotation Hooke dumbbell model admits a global weak solution provided the initial data under different integrable conditions. Moreover, we obtain optimal time decay rate in $L^2$ for the weak solutions obtained by the Fourier splitting method.