Large time behavior for the Oldroyd-B model

This paper studies the optimal time decay to the Oldroyd-B model in \({\mathbb {R}}^d\) ${\mathbb{R}}^d$ ( \(d\ge 2\) $d\ge 2$ ). By appealing to a refined pure energy method, we prove the lower and upper bounds of decay estimates of the global solutions. In particular, the lower bound of decay estimates is established by fully exploiting the structure of this system and by introducing a new combined quantity.