Global regularity for the 2D Oldroyd-B model in the corotational case

TL;DR

This paper proves the global regularity of the 2D Oldroyd-B model with fractional dissipation in the corotational case, using novel methods that differ from previous iterative approaches, and extends results to critical and supercritical cases.

Contribution

It establishes the global smooth solutions for the 2D Oldroyd-B model with fractional dissipation in the corotational case, including critical and supercritical regimes, using new analytical techniques.

Findings

Global regularity for fractional dissipation with any lpha>0.

New methods avoiding iterative approaches used in prior work.

Extension of results to critical and supercritical cases.

Abstract

This paper is dedicated to the Oldroyd-B model with fractional dissipation $(-\Delta)^{\alpha}\tau$ for any $\alpha>0$. We establish the global smooth solutions to the Oldroyd-B model in the corotational case with arbitrarily small fractional powers of the Laplacian in two spatial dimensions. The methods described here are quite different from the tedious iterative approach used in recent paper \cite{XY}. Moreover, in the Appendix we provide some a priori estimates to the Oldroyd-B model in the critical case which may be useful and of interest for future improvement. Finally, the global regularity to to the Oldroyd-B model in the corotational case with $-\Delta u$ replaced by $(-\Delta)^{\gamma}u$ for $\gamma>1$ are also collected in the Appendix. Therefore our result is more closer to the resolution of the well-known global regularity issue on the critical 2D Oldroyd-B model.