Global strong solutions and large time behavior to a micro-macro model for compressible polymeric fluids near equilibrium

TL;DR

This paper establishes the existence of global strong solutions and their long-term decay behavior for a coupled micro-macro model of compressible polymeric fluids near equilibrium, using advanced Fourier analysis techniques.

Contribution

It proves the global existence of strong solutions for the model and introduces a new Fourier estimate to analyze decay rates of derivatives.

Findings

Existence of unique global strong solutions near equilibrium for dimensions d≥2.

Development of a new critical Fourier estimate for decay analysis in d≥3.

Quantitative decay rates for all order spatial derivatives over time.

Abstract

In this paper, we mainly study the global strong solutions and its long time decay rates of all order spatial derivatives to a micro-macro model for compressible polymeric fluids with small initial data. This model is a coupling of isentropic compressible Navier-Stokes equations with a nonlinear Fokker-Planck equation. We first prove that the micro-macro model admits a unique global strong solution provided the initial data are close to equilibrium state for $d\geq2$. Moreover, for $d\geq3$, we also show a new critical Fourier estimation that allow us to give the long time decay rates of $L^2$ norm for all order spatial derivatives.