On $L^1$ estimates of solutions of compressible viscoelastic system

TL;DR

This paper analyzes the long-term behavior of solutions to a 3D compressible viscoelastic system, demonstrating that small initial perturbations lead to solutions growing at a rate of t^{1/2} in L^1, driven by diffusion wave phenomena.

Contribution

It provides the first detailed analysis of the large-time behavior of solutions to the compressible viscoelastic system in three dimensions, highlighting the diffusion wave effects.

Findings

Solutions grow at rate t^{1/2} in L^1 norm.

Diffusion wave phenomena dominate the long-term behavior.

Small initial perturbations lead to predictable growth patterns.

Abstract

We consider the large time behavior of solutions of compressible viscoelastic system around a motionless state in a three-dimensional whole space. We show that if the initial preturbation belongs to $W^{2,1}$, and is sufficiently small in $H^4\cap L^1$, the solutions grow in time at the same rate as $t^{\frac{1}{2}}$ in $L^1$ due to diffusion wave phenomena of the system caused by interaction between sound wave, viscous diffusion and elastic wave.