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

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.