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Ill-posedness issue on the Oldroyd-B model in the critical Besov spaces

Part of: Equations of mathematical physics and other areas of application Partial differential equations Compressible fluids and gas dynamics, general Incompressible viscous fluids

Published online by Cambridge University Press:  29 August 2025

Jinlu Li ,
Yanghai Yu  and
Weipeng Zhu
Jinlu Li
Affiliation:
School of Mathematics and Computer Sciences, Gannan Normal University, Ganzhou 341000, China (lijinlu@gnnu.edu.cn)
Yanghai Yu
Affiliation:
School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, China (yuyanghai214@sina.com)
Weipeng Zhu
Affiliation:
School of Mathematics, Foshan University, Foshan, Guangdong 528000, China (mathzwp2010@163.com)
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Abstract

It was proved in [11, J. Funct. Anal., 2020] that the Cauchy problem for some Oldroyd-B model is well-posed in $\dot{B}^{d/p-1}_{p,1}(\mathbb{R}^d) \times \dot{B}^{d/p}_{p,1}(\mathbb{R}^d)$ with $1\leq p \lt 2d$. In this paper, we prove that the Cauchy problem for the same Oldroyd-B model is ill-posed in $\dot{B}^{d/p-1}_{p,r}(\mathbb{R}^d) \times \dot{B}^{d/p}_{p,r}(\mathbb{R}^d)$ with $1\leq p\leq \infty$ and $1 \lt r\leq\infty$ due to the lack of continuous dependence of the solution.

Keywords

Oldroyd-B modelIll-posednessBesov spacesCauchy problemcontinuous dependence

MSC classification

Primary: 35Q35: PDEs in connection with fluid mechanics
Secondary: 35B65: Smoothness and regularity of solutions 76D05: Navier-Stokes equations 76N10: Existence, uniqueness, and regularity theory

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 20
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh.

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