## Prandtl flow past a flat plate - direct method

We introduce the following class of nonlinear problems, which is considered in detail in Oleinik and Samokhin (1999).

Problem Class 12.1. Oleinik and Samokhin nonlinear system in two dimensions.

Find u e = (ue,ve) such that for all (x,y) G fl

For a typical problem from Problem Class 12.1, with sufficiently smooth data g and sufficient compatibility, the existence and uniqueness of a solution ut can

(12.1a)

(12.1b) (12.1c) (12.Id)

be established by the techniques in Oleinik and Samokhin (1999). We denote by T7 that part of the boundary on which boundary conditions must be specified. From the parabolic nature of the problem, and the fact that the variable x is the time-like variable, it is clear that T' = Tl U T t U Tb-

A particular problem from this class is the following

where up is the exact solution of (Pp). Since the boundary conditions in (P£) are the appropriate values of up on T', it follows that on the finite rectangle Q the solution u£ of (P£) is equal to the solution up of (Pp).