In this volume we present lectures by M. C. Lopes concerning the boundary layers of incompressible fluid flow; by C. J. Xu on the micro-local analysis and its applications to the regularities of kinetic equations; by Y. X. Zheng on the weak solutions of variational wave equation from liquid crystals; and by P. Zhang and Z. F. Zhang on the free boundary problem of Euler equations. In addition, we also included lectures by F. Nier on the hypoellipticity of Fokker-Planck operator and Witten-Laplace operator. We hope that these lectures may serve as valuable references, providing up-to-date descriptions of current developments in various related research topics, that will benefit young researchers or graduate students.