Diffusion wave phenomena and optimal time decay for incompressible viscoelastic flows

Motivated by the work of D. Hoff and K. Zumbrun (Indiana Univ. Math. J. 44: 603-676, 1995), we investigate the diffusion wave phenomena in three-dimensional incompressible viscoelastic flows. By employing the representation formula of the wave equation and the stationary phase methods on the sphere ${\mathbb{S}}^{d-1}$, we establish $L^p$ decay estimates for the solution over the whole range $1\le p\le \mathrm{\infty }$, which reveals the hyperbolic nature of the incompressible viscoelastic flows.