On Global-in-x Stability of Blasius Profiles

Abstract

We characterize the well known self-similar Blasius profiles, $[{\bar{u}}, {\bar{v}}]$, as downstream attractors to solutions [u, v] to the 2D, stationary Prandtl system. It was established in Serrin (Proc R Soc Lond A 299:491–507, 1967) using maximum principle techniques that $\Vert u - {\bar{u}}\Vert _{L^\infty _y} \rightarrow 0$ as $x \rightarrow \infty$. In the case of localized data near Blasius, this paper provides an energy based proof of asymptotic stability. Central to our analysis is a new weighted “quotient estimate” which couples with a higher order, nonlinear energy cascade. Similar quotient estimates have played a crucial role in establishing the validity of the inviscid Prandtl layer expansion in Guo and Iyer (Validity of steady Prandtl layer expansions. arXiv:1805.05891 2018).