Optimal time-decay estimates for an Oldroyd-B model with zero viscosity

TL;DR

This paper establishes optimal time-decay rates for solutions to a 3D diffusive Oldroyd-B model, revealing that these rates are independent of viscosity or diffusion, unlike in Navier-Stokes equations.

Contribution

It derives the first optimal decay estimates for the Oldroyd-B model with zero viscosity, highlighting a unique independence from fluid viscosity.

Findings

Optimal decay rates are independent of viscosity.

Decay estimates are valid for small initial data with bounded Fourier transform.

The results provide new insights into the long-term behavior of viscoelastic fluids.

Abstract

In this work, we consider the Cauchy problem for a diffusive Oldroyd-B model in three dimensions. Some optimal time-decay rates of the solutions are derived via analysis of upper and lower time-decay estimates provided that the initial data are small and that the absolute value of Fourier transform of the initial velocity is bounded below away from zero in a low-frequency region. It is worth noticing that the optimal rates are independent of the fluid viscosity or the diffusive coefficient, which is a different phenomenon from that for incompressible Navier-Stokes equations.