| In this thesis we are concerned with partial differential equations arising from boundary layer problems, namely, hydrostatic Euler equations and Prandtl equations. After a brief survey of the current progress of the hydrostatic Euler equations, we establish the local existence and uniqueness of Hs solutions under the local Rayleigh condition. The proof is based on new weighted Hs estimates. Moreover, we provide a finite time blowup result as well. For the Prandtl equations, we first give a brief introduction of the known existence and ill-posedness results. Then, under Oleinik's monotonicity assumption, we prove the local existence of weak solutions to the Prandtl equations by the viscous-splitting algorithm. |
