Global well-posedness to the subcritical Oldroyd-B type models in 2D

We prove the global well-posedness to the 2D Oldroyd-B type models with $\nu \Lambda^{2\alpha}u$and$\eta\Lambda^{2\beta}\tau$satisfying$(i)\ \alpha>1, \eta=0$or$(ii)\ \alpha=1,\ \beta>0$\mathrm{.}Byestablishingthegradientestimateof$u$,$\tau$and$L^\infty$boundof${\rm curl u+\Lambda^{-2}curldiv \tau}$, Elgidi-Rousset (Commun. Pure Appl. Math. online, 2015) obtained the global well-posedness for the case $\nu =0$, $\beta =1$. However, for the cases $(i)$ and $(ii)$, it is difficult to improve the regularity of $u$ and $\tau$ directly, especially when $\alpha \to 1^{+}$ in case $(i)$ and $\beta \to 0^{+}$ in case $(ii)$. To overcome this difficulty, we exploit a new structure of the equations coming from the dissipation and coupled term. Then we prove the global well-posedness to these cases by energy method which brings us closer to the more interesting case $\alpha =1$, $\eta =0$.