Gevrey class smoothing effect for the Prandtl equation
Résumé
It is well know that the non linear Prandtl boundary layer equation is instable, and the well-posedness of the Cauchy problem in Sobolev space is an open problem. Recently, under the Oleinik's monotonicity assumption for the initial datum, \cite{awxy} have proved the local well-posedness of Cauchy problem in Sobolev space. In this work, we study the Gevrey smoothing effects of the local solution obtained in \cite{awxy}. We prove that the Soblev's class solution is belongs to some Gevrey class with respect to tangential variables at positive time. We get also the precise Gevery norms estimate with respect to time variable. This qualitative study of the solution for Prandtl's equation in Gevery class can help us to understand the Prandtl boundary layer theory which is justified in analytic frame.
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