This paper is dedicated to investigating large time decay of weak solutions for the 2D Oldroyd-B model of non-Newtonian flows. Due to the inconsistencies of the dissipative effect between Laplacian velocity viscosity and linear stress damping, either the classic Fourier splitting methods or Kato’s methods for both velocity $u$ and stress tensor $\tau$ are not available directly. Based on a new observation on the structure of Oldroyd-B model instead of the complex spectral analysis of linear problem, we explore the low frequency effect for weak solutions which allows us to develop the generalized Fourier splitting technique. More precisely, we first examine that the weak solutions with $L^2$ initial data have the non-uniformly decay ${\Vert u\left(t\right)\Vert }_{L^2\left({\mathbb{R}}^2\right)}+{\Vert \tau \left(t\right)\Vert }_{L^2\left({\mathbb{R}}^2\right)}\to 0,t\to \infty .$ Furthermore, we obtain the optimal decay rates of the weak solutions ${\Vert \tau \Vert }_{L^2\left({\mathbb{R}}^2\right)}+{\Vert u\Vert }_{L^2\left({\mathbb{R}}^2\right)}\le C{(1+t)}^{-\frac{1}{2}}$ when $(u_0,{\tau }_0)\in L^2\left({\mathbb{R}}^2\right)\cap L^1\left({\mathbb{R}}^2\right)$. Our methods here are also available for the time decay issue of the complex fluid flows with partial dissipation.

本文致力于研究非牛顿流的二维Oldroyd-B模型的弱解的长时间衰减。由于拉普拉斯速度黏度和线性应力阻尼之间的耗散效应不一致，因此对于这两种速度，经典的傅里叶分裂方法或加藤方法$ü$ 和应力张量 $\tau$无法直接使用。基于对Oldroyd-B模型结构的新观察，而不是对线性问题进行复杂的频谱分析，我们探索了弱解的低频效应，这使我们能够开发广义傅里叶分裂技术。更准确地说，我们首先检查${\mathrm{大号}}^2$ 初始数据具有非均匀衰减 ${\Vert ü（Ť）\Vert }_{{\mathrm{大号}}^2（{\mathbb{[R}}^2）}+{\Vert \tau （Ť）\Vert }_{{\mathrm{大号}}^2（{\mathbb{[R}}^2）}\to 0，Ť\to \infty 。$ 此外，我们获得了弱解的最佳衰减率 ${\Vert \tau \Vert }_{{\mathrm{大号}}^2（{\mathbb{[R}}^2）}+{\Vert ü\Vert }_{{\mathrm{大号}}^2（{\mathbb{[R}}^2）}\le C{（\mathrm{1个}+Ť）}^{-\frac{\mathrm{1个}}{2}}$ 什么时候 $（{ü}_0，{\tau }_0）\in {\mathrm{大号}}^2（{\mathbb{[R}}^2）\cap {\mathrm{大号}}^{\mathrm{1个}}（{\mathbb{[R}}^2）$。我们的方法也可用于部分耗散的复杂流体流的时间衰减问题。