High Reynolds number and high Weissenberg number Oldroyd-B model with dissipation

Splash singularities for a general Oldroyd model with finite Weissenberg number

In this paper we study a 2D Oldroyd free-boundary model which describes the evolution of a viscoelastic fluid. We prove existence of splash singularities, namely points where the boundary remains smooth but self-intersects. This paper extends the previous results obtained for infinite Weissenberg number to the case of any finite Weissenberg number. The main difficulty of this paper is due to the non linear balance law of the elastic tensor which cannot be reduced, as in the case of infinite Weissenberg, to the transport equations for the deformation gradient. Our strategy in accurate local exi…

二维不可压缩无粘Oldroyd-B方程：不恰当性、长时间存在性及高Weissenberg数极限

2D incompressible inviscid Oldroyd-B equations: ill-posedness, long time existence, and high Weissenberg number limit

本文研究二维不可压缩无粘Oldroyd-B方程在高Weissenberg数极限下的行为，证明了其线性和非线性不恰当性，并通过Voigt正则化稳定系统，建立了长时间well-posedness和均匀能量估计，最终显示解收敛于Navier-Stokes方程解。

Mathematical analysis of the stationary Oldroyd model with diffusive stress

We present the mathematical analysis of the stationary Oldroyd model with diffusive stress: existence and uniqueness of weak solutions is shown if the source terms are small enough or if the Reynolds and Weissenberg numbers are small enough. Besides, in the corotational model, this condition on the data can be relaxed for the existence result. Finally, strong solutions are obtained with additional regularity on the data.

二维蠕动流区域内应力扩散型Oldroyd-B模型初边值问题强解的整体适定性

Global well-posedness of strong solutions to the initial-boundary value problem for a two-dimensional stress-diffusive Oldroyd-B model in the creeping flow regime

本文证明了二维蠕动流区域内应力扩散型Oldroyd-B模型的初边值问题，在构形张量非负性条件下，对任意大的H¹初始聚合物应力均存在唯一整体强解，还揭示了零雷诺数区域的速度场正则性特征。

刚性粒子悬浮液多尺度粘弹性模型的流体动力学极限

Hydrodynamic limit of multiscale viscoelastic models for rigid particle suspensions

本文研究刚性棒状粒子悬浮液多尺度粘弹性模型在小 Weissenberg 数下的流体动力学极限，证明有序流体模型作为渐近近似的有效性，并建立了非线性流体模型的新适定性与正则性结果。

A Beale-Kato-Majda breakdown criterion for an Oldroyd-B fluid in the creeping flow regime

We derive a criterion for the breakdown of solutions to the Oldroyd-B model in $\R^3$ in the limit of zero Reynolds number (creeping flow). If the initial stress field is in the Sobolev space $H^m$, $m> 5/2$, then either a unique solution exists within this space indefinitely, or, at the time where the solution breaks down, the time integral of the $L^\infty$-norm of the stress tensor must diverge. This result is analogous to the celebrated Beale-Kato-Majda breakdown criterion for the inviscid Eluer equations of incompressible fluids.