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.

This paper studies the regularity results of classical solutions to the two-dimensional critical Oldroyd-B model in the corotational case. The critical case refers to the full Laplacian dissipation in the velocity or the full Laplacian dissipation in the non-Newtonian part of the stress tensor. Whether or not their classical solutions develop finite time singularities is a difficult problem and remains open. The object of this paper is two-fold. Firstly, we establish the global regularity result to the case when the critical case occurs in the velocity and a logarithmic dissipation occurs in the non-Newtonian part of the stress tensor. Secondly, when the critical case occurs in the non-Newtonian part of the stress tensor, we first present many interesting global a priori bounds, then establish a conditional global regularity in terms of the non-Newtonian part of the stress tensor. This criterion comes naturally from our approach to obtain a global L∞-bound for the vorticity ω.