Title: The Cauchy problem for an inviscid and non-diffusive Oldroyd-B model in two dimensions Show Chinese title

Title: 二维无粘性非扩散Oldroyd-B模型的柯西问题

Abstract: A two-dimensional inviscid and diffusive Oldroyd-B model was investigated by [T. M. Elgindi, F. Rousset, Commun. Pure Appl. Math. 68 (2015), 2005--2021] where the global existence and uniqueness of the strong solution were established for arbitrarily large initial data. As pointed out by [A. V. Bhave, R. C. Armstrong, R. A. Brown, J. Chem. Phys. 95(1991), 2988--3000], 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 via 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. Show Chinese abstract

Abstract: 二维无粘性和扩散的Oldroyd-B模型被[T. M. Elgindi, F. Rousset, Commun. Pure Appl. Math. 68 (2015), 2005--2021]研究，其中对于任意大的初始数据，建立了强解的全局存在性和唯一性。如[ A. V. Bhave, R. C. Armstrong, R. A. Brown, J. Chem. Phys. 95(1991), 2988--3000]所指出的，扩散系数明显小于其他效应，因此研究非扩散模型是有趣的。在本工作中，我们通过推导一些均匀正则性估计并取消失扩散极限，得到了小初始数据下非扩散模型强解的全局时间存在性和唯一性。此外，研究了解的大时间行为，并获得了每个阶数的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正则性效应以及二维设置中较慢的衰减上。采用谱分析和傅里叶分裂方法的组合。