Global wellposedness and large time behavior of solutions to the $n$-dimensional compressible Oldroyd-B model

TL;DR

This paper establishes the global existence and decay rates of solutions to the compressible Oldroyd-B model using harmonic analysis, allowing for initial data with low regularity and vanishing polymer density.

Contribution

It proves global well-posedness and decay rates for the model with low regularity initial data, extending previous results by accommodating vanishing polymer density.

Findings

Global well-posedness for small initial data in low regularity spaces.

Optimal decay rates under low frequency initial data conditions.

Polymer density can vanish, and stress tensor approaches zero equilibrium.

Abstract

The purpose of this work is to study the global wellposedness and large time behavior results of strong solutions for the compressible Oldroyd-B model derived by Barrett, Lu, S\"uli (Commun. Math. Sci., 15, 1265--1323, 2017). Exploiting the Harmonic analysis tools (especially Littlewood-Paley theory), we first study the global well-posedness of the model with small initial data in spaces with low regularity. Then, under a suitable condition involving only the low frequency of the initial data, we also obtain the optimal decay rates of the solutions. Compared with the result by Wang and Wen (Math. Models Methods Appl. Sci., 30, 139--179, 2020), the polymer number density is allowed to vanish and the stress tensor isnear zero equilibrium.