Diffusion wave phenomena and optimal time decay for incompressible viscoelastic flows

TL;DR

This paper studies the diffusion wave phenomena in 3D incompressible viscoelastic flows, establishing decay estimates that highlight their hyperbolic nature using advanced mathematical techniques.

Contribution

It introduces new $L^p$ decay estimates for these flows, revealing their hyperbolic characteristics through representation formulas and stationary phase methods.

Findings

Established $L^p$ decay estimates for solutions.

Revealed the hyperbolic nature of the flows.

Extended previous work on diffusion phenomena.

Abstract

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 $1\leq p \leq \infty$, which reveals the hyperbolic nature of the incompressible viscoelastic flows.