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 $νΛ^{2α}u$ and $ηΛ^{2β}τ$ satisfying $(i)\ α>1, η=0$ or $(ii)\ α=1,\ β>0$. By establishing the gradient estimate of $u$, $τ$ and $L^\infty$ bound of ${\rm curl u+Λ^{-2}curldiv τ}$, Elgidi-Rousset (Commun. Pure Appl. Math. online, 2015) obtained the global well-posedness for the case $ν=0$, $β=1$. However, for the cases $(i)$ and $(ii)$, it is difficult to improve the regularity of $u$ and $τ$ directly, especially when $α\rightarrow 1^{+}$ in case $(i)$ and $β\rightarrow 0^{+}$ 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 $α=1$, $η=0$.

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