We consider the Cauchy problem for incompressible viscoelastic fluids in thewhole space $\mathbb{R}^d$ ($d=2,3$). By introducing a new decomposition viaHelmholtz's projections, we first provide an alternative proof on the existenceof global smooth solutions near equilibrium. Then under additional assumptionsthat the initial data belong to $L^1$ and their Fourier modes do not degenerateat low frequencies, we obtain the optimal $L^2$ decay rates for the globalsmooth solutions and their spatial derivatives. At last, we establish theweak-strong uniqueness property in the class of finite energy weak solutionsfor the incompressible viscoelastic system.
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机译：我们考虑整个空间$ \ mathbb {R} ^ d $（$ d = 2,3 $）中不可压缩粘弹性流体的柯西问题。通过通过亥姆霍兹的投影引入新的分解，我们首先提供关于平衡附近整体光滑解的存在性的替代证明。然后在额外的假设下，即初始数据属于$ L ^ 1 $并且它们的傅里叶模式不会在低频下退化，我们获得了全局平滑解及其空间导数的最佳$ L ^ 2 $衰减率。最后，在不可压缩粘弹性系统的有限能量弱解类中建立了弱强唯一性。
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