A two-dimensional inviscid and diffusive Oldroyd-B model was investigated by Elgindi and Rousset (2015) where the global existence and uniqueness of the strong solution were established for arbitrarily large initial data. As pointed out by Bhave et al. (1991), since the diffusion coefficient is significantly smaller than other effects, it is interesting to study the non-diffusive model. In the present work, we obtain the global-in-time existence and uniqueness of the strong solution to the non-diffusive model with small initial data by deriving some uniform regularity estimates and taking vanishing diffusion limits. In addition, the large time behavior of the solution is studied and the optimal time-decay rates for each order of spatial derivatives are obtained. The main challenges focus on the lack of dissipation and regularity effects of the system and on the slower decay in the two-dimensional settings. A combination of the spectral analysis and the Fourier splitting method is adopted.

中文翻译：

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二维无粘非扩散 Oldroyd-B 模型的柯西问题

Elgindi 和 Rousset (2015) 研究了二维无粘扩散 Oldroyd-B 模型，其中针对任意大的初始数据建立了强解的全局存在性和唯一性。正如 Bhave 等人所指出的。(1991)，由于扩散系数明显小于其他效应，因此研究非扩散模型很有趣。在目前的工作中，我们通过推导一些一致的正则性估计并采用消失的扩散极限，获得了具有小初始数据的非扩散模型的强解的全局时间存在性和唯一性。此外，还研究了解的大时间行为，并获得了每个阶空间导数的最佳时间衰减率。主要挑战集中在系统缺乏耗散和规律性效应以及二维环境中衰减较慢。采用谱分析和傅里叶分裂方法相结合的方法。