二维 Prandtl 方程单调剪切流在 Sobolev 空间中的整体稳定性

Given initial data that is a small perturbation to the initial value of some shear flow, which is monotonic with respect to the y variable and decays sufficiently fast at large time, we prove the global well-posedness of the two-dimensional Prandtl system in Sobolev spaces. The main idea of the proof is to combine the non-linear cancelation property that was proposed by Masmoudi and Wong (2015) with the good quantity that was introduced by Paicu and Zhang (2021) which leads to a faster large-time decay estimate of the solution. The reason why we add a force term in the shear flow equation is that there does not exist any monotonic shear flow with such large-time decay rates as required by our assumption.