Well-posedness in Gevrey space for the Prandtl equations with non-degenerate critical points
Well-posedness in Gevrey space for the Prandtl equations with non-degenerate critical points
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发表时间：
2016-09
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通讯作者：
Wei-Xi Li;Tong Yang
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作者：
Wei-Xi Li;Tong Yang
本文研究了具有非退化临界点的Prandtl系统。对于[3/2，2]中的任意指标$\sigma，$我们得到了切向变量Gevrey类$G^\sigma$和法向变量Sobolev类空间中的局部时间适定性，从而不需要切向速度的单调性条件来克服切向导数的损失.这回答了D. G'e}rard-Varet和N. Masmoudi [{\it Ann. Sci. \'{E}c.诺姆Sup\'{e}r}. (4)48（2015），no. 6，1273-1325]，其中解决了$\sigma=7/4$的情况。
In the paper, we study the Prandtl system with initial data admitting non-degenerate critical points. For any index $\sigma\in[3/2, 2],$ we obtain the local in time well-posedness in the space of Gevrey class $G^\sigma$ in the tangential variable and Sobolev class in the normal variable so that the monotonicity condition on the tangential velocity is not needed to overcome the loss of tangential derivative. This answers the open question raised in the paper of D. G\'{e}rard-Varet and N. Masmoudi [{\it Ann. Sci. \'{E}c. Norm. Sup\'{e}r}. (4) 48 (2015), no. 6, 1273-1325], in which the case $\sigma=7/4$ is solved.