GEVREY CLASS SMOOTHING EFFECT FOR THE PRANDTL EQUATION

It is well known that the Prandtl boundary layer equation is unstable for general initial data, and is well-posed in Sobolev space for monotonic initial data. Recently, under the Oleinik's monotonicity assumption for the initial datum, R. Alexandre, Y. Wang, C.-J. Xu, and T. Yang [J. Amer. Math. Soc., 28 (2015) pp. 745-784] recovered the local well-posedness of the Cauchy problem in Sobolev space by virtue of an energy method (see also N. Masmoudi and T. K. Wong [Comm. Pure Appl. Math., 68 (2015), pp. 1683-1741.]). In this work, we study the Gevrey smoothing effects of the local solution obtained in R. Alexandre, Y. Wang, C.-J. Xu, and T. Yang [J. Amer. Math. Soc., 28 (2015) pp. 745-784]. We prove that the Sobolev's class solution belongs to some Gevrey class with respect to tangential variables at any positive time.