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

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.Comment: 23 pages, Submitted August 201

Steady flows of an Oldroyd fluid with threshold slip

“…Starting with the pioneering works of Renardy [35] and Guillopé & Saut [20], mathematical models of viscoelastic fluids of Oldroyd type have been studied by many authors. We mention here only the papers [2,3,5,9,10,12,13,14,15,16,17,18,21,22,26,28,30,39,40,41]. A detailed analysis of different problems and results related to the Oldroyd model and other similar non-Newtonian models can be found in the review article [37].…”

“…We shall construct a weak solution of the original boundary value problem by passing to the limit n → ∞. In view of the estimates (41) and (42), we can assume without loss of generality that, for some v * ∈ X(Ω,…”

Mathematical study of a single leukocyte in microchannel flow

“…The existence of local strong solutions for the Oldroyd-B model was proved in [2,12,13]. The reader can refer to [9,10,15,23,24,28,33] for other interesting results on the Oldroyd-B and related models. In all the above references, the density is considered to be constant.…”

An Osgood type regularity criterion for the 2D Oldroyd‐B model

“…The well-posedness for the Oldroyd-B model has been studied extensively (see previous studies 2,[4][5][6][7][8][9][10] ). Chemin and Masmoudi 2 stated the existence of local solution for (1.1) in H s with s > d 2 .…”