随着现代材料科学的发展，复杂流体的Oldroyd-B模型越来越受到国内外数学工作者的重视与关注。由于非线性、耦合性以及可能出现的退化或奇性，关于Oldroyd-B模型的数学理论研究具有很大的困难。本项目将在流体力学方程组已有的理论基础上，深入挖掘耦合方程的内蕴结构和非线性结构，运用各类方程的数学理论和分析技巧，寻求新方法和新估计，深入探讨复杂流体动力学模型的适定性理论。本项目将重点研究带张量扩散的三维可压缩Oldroyd-B模型：（1）建立初(边)值问题大初值整体弱解的存在性理论；（2）研究大振荡小初值整体强解的存在性、正则性和长时间性质；（3）探讨小参数极限行为以及边界层/初始层现象。申请人希望根据前期工作经验，结合椭圆/抛物/双曲型方程的数学理论、调和分析技巧、补偿列紧原理等, 在上述问题的研究中取得进展。

With the development of modern materials science, the Oldroyd-B model of complex fluids has attracted more and more attention of mathematicians. From the mathematical point of view, the model is the coupling of the Navier-Stokes equations and the stress tensor equations. Because of the nonlinearity, coupling and possible degeneration or singularity, the mathematical analysis of the Oldroyd-B model is very challenging. Based on the existing literature of fluid mechanics equations, we aim to study the intrinsic and nonlinear structure of coupled equations, and develop new methods and new estimates to explore the mathematical theory of complex fluid dynamics models by applying the techniques of various kinds of equations and the modern analytical tools. This project will focus on the three-dimensional compressible Oldroyd-B model with tensor diffusion: (1) the existence theory of global weak solutions with large data to the initial (boundary) value problems; (2) the existence, regularity and asymptotic behavior of global strong solutions with large oscillations; (3) the vanishing limit and the boundary layer/initial layer phenomenon of small physical parameters. We hope to make some important progress in the mathematical analysis of the above problems by combining the regularity theory of elliptic/parabolic/hyperbolic equations, the techniques of harmonic analysis, the method of compensated compactness, and so on.