Almost Global Existence for the Prandtl Boundary Layer Equations

We consider the Prandtl boundary layer equations on the half plane, with initial datum that lies in a weighted H¹ space with respect to the normal variable, and is real-analytic with respect to the tangential variable. The boundary trace of the horizontal Euler flow is taken to be a constant. We prove that if the Prandtl datum lies within ({varepsilon}) of a stable profile, then the unique solution of the Cauchy problem can be extended at least up to time ({T_{varepsilon} geqq {rm exp}(varepsilon^{-1} / {rm log}(varepsilon^{-1}))}).