### Abstract

Consider the Cauchy problem for the incompressible Oldroyd-B model in R
^{3}
. For the case a = 0, global existence results for weak solutions were derived by Lions and Masmoudi (2000 Chin. Ann. Math. B 21 13146), allowing the initial data to be arbitrarily large, whereas it is not known whether this assertion is also true for a -= 0. In this article, time decay estimates for weak solutions subject to arbitrary large data are given for the case a = 0. Furthermore, timedecay estimates are also given for strong solutions for a = 0, however, for small initial data. The decay estimates obtained are of the form that the kth order derivatives in L
^{2}
decay as (1 + t)?34 ?k2 for k = 0, 1, 2 as t → ∞. Note that the coupling constant ω does not need to be small throughout this paper.

Original language | English |
---|---|

Pages (from-to) | 833-852 |

Number of pages | 20 |

Journal | Nonlinearity |

Volume | 32 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2019 Jan 29 |

Externally published | Yes |

### Fingerprint

### Keywords

- Oldroyd-B equation
- optimal decay rates
- weak solution

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)
- Applied Mathematics

### Cite this

^{3}

*Nonlinearity*,

*32*(3), 833-852. https://doi.org/10.1088/1361-6544/aaeec7

**
Optimal decay rates for solutions to the incompressible Oldroyd-B model in R
^{3}
.** / Hieber, Matthias Georg; Wen, Huanyao; Zi, Ruizhao.

Research output: Contribution to journal › Article

^{3}'

*Nonlinearity*, vol. 32, no. 3, pp. 833-852. https://doi.org/10.1088/1361-6544/aaeec7

^{3}Nonlinearity. 2019 Jan 29;32(3):833-852. https://doi.org/10.1088/1361-6544/aaeec7

}

TY - JOUR

T1 - Optimal decay rates for solutions to the incompressible Oldroyd-B model in R 3

AU - Hieber, Matthias Georg

AU - Wen, Huanyao

AU - Zi, Ruizhao

PY - 2019/1/29

Y1 - 2019/1/29

N2 - Consider the Cauchy problem for the incompressible Oldroyd-B model in R 3 . For the case a = 0, global existence results for weak solutions were derived by Lions and Masmoudi (2000 Chin. Ann. Math. B 21 13146), allowing the initial data to be arbitrarily large, whereas it is not known whether this assertion is also true for a -= 0. In this article, time decay estimates for weak solutions subject to arbitrary large data are given for the case a = 0. Furthermore, timedecay estimates are also given for strong solutions for a = 0, however, for small initial data. The decay estimates obtained are of the form that the kth order derivatives in L 2 decay as (1 + t)?34 ?k2 for k = 0, 1, 2 as t → ∞. Note that the coupling constant ω does not need to be small throughout this paper.

AB - Consider the Cauchy problem for the incompressible Oldroyd-B model in R 3 . For the case a = 0, global existence results for weak solutions were derived by Lions and Masmoudi (2000 Chin. Ann. Math. B 21 13146), allowing the initial data to be arbitrarily large, whereas it is not known whether this assertion is also true for a -= 0. In this article, time decay estimates for weak solutions subject to arbitrary large data are given for the case a = 0. Furthermore, timedecay estimates are also given for strong solutions for a = 0, however, for small initial data. The decay estimates obtained are of the form that the kth order derivatives in L 2 decay as (1 + t)?34 ?k2 for k = 0, 1, 2 as t → ∞. Note that the coupling constant ω does not need to be small throughout this paper.

KW - Oldroyd-B equation

KW - optimal decay rates

KW - weak solution

UR - http://www.scopus.com/inward/record.url?scp=85062541435&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85062541435&partnerID=8YFLogxK

U2 - 10.1088/1361-6544/aaeec7

DO - 10.1088/1361-6544/aaeec7

M3 - Article

VL - 32

SP - 833

EP - 852

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 3

ER -