The continuation problem for the Prandtl boundary layer (Q5935950)

The author shows that there exists a class of initial value problems, for which the boundary layer can be continued up to the stagnation point. The Prandtl system of boundary layer equations \[ u\partial u/\partial x + v\partial u/\partial y = \nu\partial^2 u/ \partial y^2 - dp/dx, \qquad \partial u/\partial x + \partial v/\partial y = 0 \] is considered for \(x>0\), \(y>0\), with the boundary conditions \[ u|_{x=0}=u_0(y),\quad u|_{y=0}=0,\quad v|_{y=o}=0, \] and \(u(x,y) \to U(x)\) as \(y \to \infty\) uniformly with respect to \(x\). The pressure \(p(x)\) is related to the velocity \(U(x)\) by the formula \(2p(x)+U^2(x)=\)const (the Bernoulli integral).