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

Abstract

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

Version accepted for publication in Proceedings of the Royal Society of Edinburgh-Section A: Mathematics

References in corpus (2)

• Global Solutions for Incompressible Viscoelastic Fluids
• Sharp ill-posedness for the non-resistive MHD equations in Sobolev spaces