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

TL;DR

This paper proves the global well-posedness of certain 2D Oldroyd-B models with fractional dissipation, using energy methods and new structural insights to handle challenging parameter regimes.

Contribution

It establishes global existence and uniqueness results for 2D Oldroyd-B models with fractional dissipation in cases previously difficult to analyze.

Findings

Proved global well-posedness for models with $ u eq 0$ and fractional dissipation.

Developed new structural estimates to handle critical parameter limits.

Extended the understanding of regularity for Oldroyd-B type models.

Abstract

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 $(ii)\ \alpha=1,\ \beta>0$. By establishing the gradient estimate of $u$, $\tau$ and $L^\infty$ bound of ${\rm curl u+\Lambda^{-2}curldiv \tau}$, 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 $\alpha\rightarrow 1^{+}$ in case $(i)$ and $\beta\rightarrow 0^{+}$ 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 $\alpha=1$, $\eta=0$.