The optimal time decay rates for incompressible viscoelastic fluids in critical framework

We study the Cauchy problem for multi-dimensional incompressible viscoelastic fluids in the whole space. The optimal time decay rates of strong solution constructed by Qian (2010), and Zhang (2011) in $L^{2}$-critical regularity framework are obtained for low frequencies of the data under a suitable additional condition. The proof relies on an application of Fourier analysis to a mixed parabolic-hyperbolic system, and on a refined time-weighted energy functional. As a by-product, time decay rates of $L^{q}$-$L^{r}$ type are also captured in the critical framework.