二维Prandtl方程单调剪切流在Sobolev空间中的整体稳定性  

Global stability of monotone shear ows for the 2-D Prandtl system in Sobolev spaces

作　　者：刘宁 张平 Ning Liu;Ping Zhang

机构地区：[1]中国科学院数学与系统科学研究院,北京100190 [2]中国科学院大学数学科学学院,北京100049

出　　处：《中国科学：数学》2024年第3期457-482,共26页Scientia Sinica：Mathematica

基　　金：国家重点研发计划(批准号:2021YFA1000800);国家自然科学基金(批准号:12288201和12031006)资助项目。

摘　　要：考虑初值是某个关于y变量单调且长时间衰减得充分快的剪切流附近的小扰动,本文证明二维Prandtl方程在Sobolev空间中的整体适定性.本文证明的主要思路是将Masmoudi和Wong(2015)指出的非线性对消性质与Paicu和Zhang(2021)引入的能够证明解有更快长时间衰减估计的好函数结合起来.本文中需要在剪切流满足的方程中添加一个外力项,否则可以证明不存在长时间衰减足够快的单调剪切流.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 wellposedness 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.

关 键 词：Prandtl方程 适定性 SOBOLEV空间 能量方法 

分 类 号：O175[理学—数学]