Optimal Time Decay Rate for the Compressible Viscoelastic Equations in Critical Spaces

TL;DR

This paper establishes the optimal decay rates over time for solutions to the compressible viscoelastic equations in critical Besov spaces, demonstrating improved decay estimates with low-regularity initial data.

Contribution

It introduces a novel combination of Green's matrix and energy methods to derive optimal decay rates in critical Besov spaces for compressible viscoelastic fluids.

Findings

Achieved optimal $L^{2}$-time decay rates for solutions.

Demonstrated decay rates with initial data of very low regularity.

Extended decay analysis to the critical Besov space framework.

Abstract

In this paper, we are concerned with the convergence rates of the global strong solution to constant equilibrium state for the compressible viscoelastic fluids in the whole space. We combine both analysis about Green's matrix method and energy estimate method to get optimal time decay rate in critical Besov space framework. Our result imply the optimal $L^{2}$-time decay rate and only need the initial data to be small in critical Besov space which have very low regularity compared with traditional Sobolev space.