The optimal time decay rates for the compressible Oldroyd-B model in ℝ³

Abstract

In this paper, we consider the optimal time decay rates of the higher order spatial derivatives of global solution for the three-dimensional compressible Oldroyd-B model, which was derived independently by Barrett-Lu-Süli [Commun. Math. Sci., 15(5): 1265–1323, 2017] via micro–macro analysis of the compressible Navier–Stokes–Fokker–Planck system. We show that under $H^3$ framework, the third-order spatial derivatives of global small solution of the compressible Oldroyd-B model decay at the rate $(1+t{)}^{-\frac{9}{4}}$ instead of $(1+t{)}^{-\frac{7}{4}}$ stated by Wang-Wen [Math. Models Methods Appl. Sci., 30(1): 139–179, 2020]. This decay rate is optimal in the sense that it coincides with that of solutions to heat equation.

Keywords:

• Compressible Oldroyd-B model
• optimal time decay rates
• viscoelastic fluids

2020 Mathematics Subject Classifications:

• Primary 35Q35
• Secondary 76A10
• 74H40

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Additional information

Funding

This work was supported by NNSF of China [grant no. 12201655]; NNSF of China [grant no. 11972384]; NNSF of China [grant no. 11971496]; Guangdong Basic and Applied Basic Research Foundation [grant no. 2023A1515010974]; Guangdong Science and Technology Fund [grant no. 2021B1515310001]; and National Key R$\mathrm{\&}$D Program of China [grant no. 2020YFA0712500].