Remarks on the Prandtl boundary layer

In a recent result of Gérard-Varet and Dormy (2010) [4], , they established ill-posedness for the Cauchy problem of the linearized Prandtl equation around non-monotic special solution which is independent of x and satisfies the heat equation. In Guo and Nguyen (2010) [5], and Gérard-Varet and Nguyen (2010) [6], some nonlinear ill-posedness were established with this counterexample. Then it is natural to consider the problem that does this linear ill-posedness happen whenever the non-degenerate critical points appear. In this paper, we concern the linearized Prandtl equation around general stationary solutions with non-degenerate critical points depending on x which could be considered as the time-periodic solutions and show some ill-posedness.