In 1904, Prandtl said that, in fluid of small viscosity, the behavior of fluid near the boundary is completely different from that away from the boundary. Away from the boundary part can be almost considered as ideal fluid, but the near boundary part is deeply affected by the viscous force and is described by Prandtl boundary layer equation which was firstly derived formally by Prandtl. From the mathematical point of view, the well-posedness and justification of the Prandtl boundary layer theory don’t have satisfactory theory yet. In this talk, we present some recent progress on the mathematical analysis of the Prandtl boundary layer equation. By using energy method, we study the well-posedness of Cauchy problem and the smoothness effect of solutions for Prandtl equations in Sobolev space.

[1] R. Alexandre, Y. G. Wang, C.-J. Xu and T. Yang :  Well-posedness of the Prandtl  equation in Sobolev spaces, Journal of the Amercican Mathematical Society. 28 (2015) 745-784

[2] W.-X. Li, D. Wu and C.-J. Xu, Gevery class smoothing effect for the Prandtl equation,. SIAM J. Math. Anal. 48 (2016). 1672-1726.

[3] C.-J. Xu and X. Zhang, Long time well-posdness of the Prandtl equations in Sobolev space. J. Differential Equations,. 263 (2017)  8749-8803

[4] W.-X. Li, V.-S. Ngo and C.-J. Xu, Boundary layer analysis for the fast horizontal rotating fluid,  Preprint