Optimal time-decay estimates for a diffusive Oldroyd-B model

TL;DR

This paper establishes the optimal decay rates for higher order spatial derivatives of solutions to a diffusive Oldroyd-B model, addressing an open problem in the mathematical analysis of this complex fluid dynamics system.

Contribution

It provides the first rigorous proof of optimal decay estimates for the highest-order derivatives in the diffusive Oldroyd-B model, advancing understanding of its long-term behavior.

Findings

Established optimal decay rates for higher order derivatives

Resolved an open problem in the decay estimates of the model

Analyzed low and high frequency components of solutions

Abstract

In this paper, we study the optimal time decay rates for the higher order spatial derivatives of solutions to a diffusive Oldroyd-B model. As pointed out in the Section 1.2 of Huang-Wang-Wen-Zi (J. Differential Equations 306: 456-491, 2022), how to estiblish the optimal decay estimate for the highest-order spatial derivatives of the solution to this model is still an open problem. Motivated by Wang-Wen (Sci. China Math. 65: 1199-1228, 2022), we give a positive answer to this problem via some delicate analyses on the low and high frequency parts of the solution.