Global wellposedness to the $n$-dimensional compressible Oldroyd-B model without damping mechanism

TL;DR

This paper proves the global existence and decay rates of solutions for the compressible Oldroyd-B model in multiple dimensions without damping, by exploiting intrinsic structures and new quantities, thus extending previous results.

Contribution

It introduces a novel approach to handle the lack of dissipation in the model, establishing global solutions in critical Besov spaces and deriving optimal decay rates.

Findings

Global solutions exist for small initial data.

Optimal decay rates of solutions are obtained.

Results extend previous work by removing the damping assumption.

Abstract

The Cauchy problem of the compressible Oldroyd-B model without damping mechanism in R^n$ with $n\ge2$ is considered. The lack of dissipation in density and stress tensor in the model is compensated by exploiting an intrinsic structure and introducing new quantities between density, velocity and stress tensor. Therefore, global solutions to the system with small initial data in critical Besov spaces are obtained. As a byproduct, optimal time decay rates of the solutions are derived by using an energy estimation argument. The results remain valid for the compressible viscoelastic system without the `div-curl structure assumption and thus improve those given by Hu and Wang [ J. Differential Equations, {\bf 250}, 1200--1231, 2011] and Qian and Zhang [Arch. Ration. Mech. Anal., {\bf 198}, 835--868, 2010].