Well-posedness of the Prandtl equation in Sobolev spaces
Well-posedness of the Prandtl equation in Sobolev spaces
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DOI：
10.1090/s0894-0347-2014-00813-4
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发表时间：
2012-03
影响因子：
3.9
通讯作者：
Radjesvarane Alexandre;Ya-Guang Wang;Chao-Jiang Xu;Tong Yang
中科院分区：
文献类型：
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作者：
Radjesvarane Alexandre;Ya-Guang Wang;Chao-Jiang Xu;Tong Yang
利用切向速度场单调条件下的直接能量方法代替Crocco变换，发展了一种新的方法来研究Sobolev空间中Prandtl方程的适定性理论。具体地说，我们首先在一些加权的Sobolev空间中研究了背景态切向速度按正态变量单调时的线性化Prandtl方程。然后，为了处理由于方程的退化而导致的扰动相对于背景状态的正则性损失，我们应用Nash-Moser-Hormander迭代得到了当初始数据是单调剪切流的小扰动时，非线性Prandtl方程经典解的适定性理论。
We develop a new approach to study the well-posedness theory of the Prandtl equation in Sobolev spaces by using a direct energy method under a monotonicity condition on the tangential velocity field instead of using the Crocco transformation. Precisely, we firstly investigate the linearized Prandtl equation in some weighted Sobolev spaces when the tangential velocity of the background state is monotonic in the normal variable. Then to cope with the loss of regularity of the perturbation with respect to the background state due to the degeneracy of the equation, we apply the Nash-Moser-Hormander iteration to obtain a well-posedness theory of classical solutions to the nonlinear Prandtl equation when the initial data is a small perturbation of a monotonic shear flow.