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 \cite{Serrin} that $\Vert u-\bar{u}{\Vert }_{L_y^{\infty }}\to 0$ as $x\to \infty$. Our result furthers \cite{Serrin} in the case of localized data near Blasius by establishing convergence in stronger norms and by characterizing the decay rates. Central to our analysis is a "division estimate", in turn based on the introduction of a new quantity, $\Omega$, which is globally nonnegative precisely for Blasius solutions. Coupled with an energy cascade and a new weighted Nash-type inequality, these ingredients yield convergence of $u-\bar{u}$ and $v-\bar{v}$ at the essentially the sharpest expected rates in $W^{k,p}$ norms.