Global existence and asymptotic behavior of near-affine solutions of compressible viscoelasticity

We establish the global-in-time wellposedness and asymptotic behavior of solutions to compressible viscoelasticity, in the case that the solution is a small perturbation of the affine solution. For affine solutions, the deformation gradient expands at the rate $1+t$, as $t\to \infty$. The stability of these affine solutions is constructed within small $H^2$ perturbation. A new flux is introduced to deal with the global estimate. As a byproduct, the higher decay rate of $({\nabla }^{2}w,{\nabla }^{2}E)$ is also obtained.