In this paper, we prove the global existence and the large time decay estimate of solutions to Prandtl system with small initial data, which is analytical in the tangential variable. The key ingredient used in the proof is to derive a sufficiently fast decay-in-time estimate of some weighted analytic energy estimate to a quantity, which consists of a linear combination of the tangential velocity with its primitive one, and which basically controls the evolution of the analytical radius to the solutions. Our result can be viewed as a global-in-time Cauchy–Kowalevsakya result for the Prandtl system with small analytical data, which in particular improves the previous result in Ignatova and Vicol (Arch Ration Mech Anal 220:809–848, 2016) concerning the almost global well-posedness of a two-dimensional Prandtl system.

在本文中，我们证明了具有少量初始数据的Prandtl系统解的整体存在性和较大的时间衰减估计，这是对切向变量的分析。证明中使用的关键成分是将某个加权分析能量估计值的时间上足够快的随时间衰减的估计值导出为一个量，该量由切线速度与其原始速度的线性组合组成，并且基本上控制了演化解析半径对解的影响。我们的结果可以看作是带有少量分析数据的Prandtl系统的实时Cauchy–Kowalevsakya结果，尤其是改善了Ignatova和Vicol的先前结果（Arch Ration Mech Anal 220：809–848，2016）二维Prandtl系统的几乎全局的适定性。