AI 中文总结
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。
AI 中文摘要
本文研究二维光滑有界区域内、零雷诺数（蠕动流）区域中应力扩散型Oldroyd-B系统强解的整体适定性。全文始终保留应力扩散项，将其理解为Oldroyd-B本构方程常规的质心扩散正则化项。在对应的非扩散型蠕动流情形中，现有最强结果是Kupferman、Mangoubi和Titi[Commun. Math. Sci. 6 (2008)]给出的三维柯西问题的Beale-Kato-Majda型爆破准则，即便在二维情形下整体适定性仍悬而未决。我们证明，当构形张量满足自然非负性条件时，对任意大的H¹初始聚合物应力，强解都存在唯一的整体解。该结果适用于一般光滑有界区域上的初边值问题，展示了Stokes椭圆结构、构形张量非负性的保持以及应力扩散如何结合起来封闭H¹水平的大数据估计。我们还指出了零雷诺数区域特有的一个正则性特征：速度场从椭圆型Stokes方程获得的空间正则性高于聚合物应力张量的空间正则性。
英文摘要
This paper investigates the global well-posedness of strong solutions to a stress-diffusive Oldroyd-B system in two-dimensional smooth bounded domains in the zero Reynolds number (creeping flow) regime. The stress-diffusion term is kept explicit throughout the paper and is understood as the usual center-of-mass diffusion regularization of the Oldroyd-B constitutive equation. In the corresponding non-diffusive creeping-flow setting, the strongest available result is a Beale-Kato-Majda type breakdown criterion for the three-dimensional Cauchy problem due to Kupferman, Mangoubi and Titi [Commun. Math. Sci. 6 (2008)], and global well-posedness remains open even in two dimensions. We prove the global existence and uniqueness of strong solutions for arbitrarily large H1 initial polymeric stresses satisfying the natural non-negativity condition on the conformation tensor. The result covers the initial-boundary value problem on general smooth bounded domains and shows how the Stokes elliptic structure, the preservation of the non-negativity of the conformation tensor, and the stress diffusion combine to close large-data estimates at the H1 level. We also point out a regularity feature specific to the zero Reynolds number regime: the velocity field gains higher spatial regularity from the elliptic Stokes equation than the polymeric stress tensor.
