Large time behavior in critical $L^p$ Besov spaces for compressible viscoelastic flows

TL;DR

This paper investigates the long-term decay behavior of solutions to compressible viscoelastic flows in critical Besov spaces, demonstrating optimal decay rates without small initial data assumptions using energy methods.

Contribution

It establishes optimal decay estimates for global solutions in $L^p$ critical spaces for compressible viscoelastic flows, removing the usual smallness condition on initial data.

Findings

Proves optimal decay rates similar to compressible Navier-Stokes equations.

Removes smallness assumption on initial data in low-frequency analysis.

Uses energy methods to analyze large time behavior.

Abstract

We consider the large time behavior of global strong solutions to the compressible viscoelastic flows on the whole space $\mathbb{R}^N\,(N\geq 2)$, where the system describes the elastic properties of the compressible fluid. Adding a suitable initial condition involving only the low-frequency, we prove optimal time decay estimates for the global solutions in the $L^p$ critical regularity framework, which are similar to those of the compressible Navier-Stokes equations. Our results rely on the pure energy argument, which allows us to remove the usual smallness assumption of the data in the low-frequency.