Gevrey Well-Posedness of the Hyperbolic Prandtl Equations
Gevrey Well-Posedness of the Hyperbolic Prandtl Equations
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DOI：
10.4208/cmr.2021-0104
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发表时间：
2021-12
期刊：
影响因子：
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通讯作者：
Wei-Xi Li;R. Xu
中科院分区：
文献类型：
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作者：
Wei-Xi Li;R. Xu
研究了二维和三维退化双曲型Prandtl方程，在没有任何结构假设的情况下，证明了Gevrey指数≤ 2的Gevrey适定性.与经典抛物型Prandtl方程相比，抛物型Prandtl方程的双曲性和退化性导致的导数损失，不能用经典的消除机制来克服.受抽象的Cauchy-Kowalewski定理的启发，利用双曲性，基于一个初等的L能量估计，本文给出了一个简单的证明。特别是我们的论点不涉及有效地用于经典普朗特方程的取消机制。
We study the 2D and 3D Prandtl equations of degenerate hyperbolic type, and establish without any structural assumption the Gevrey well-posedness with Gevrey index ≤ 2. Compared with the classical parabolic Prandtl equations, the loss of the derivatives, caused by the hyperbolic feature coupled with the degeneracy, can’t be overcame by virtue of the classical cancellation mechanism that developed for the parabolic counterpart. Inspired by the abstract Cauchy-Kowalewski theorem and by virtue of the hyperbolic feature, we give in this text a straightforward proof, basing on an elementary L energy estimate. In particular our argument does not involve the cancellation mechanism used efficiently for the classical Prandtl equations.