# On Global-in-$x$ Stability of Blasius Profiles

## TL;DR

This paper proves that solutions to the 2D stationary Prandtl system with localized data near Blasius profiles converge globally in $x$ to these profiles, with explicit decay rates in strong norms, using novel energy and inequality techniques.

## Contribution

It introduces a new division estimate and a quantity $ ext{ extOmega}$ to establish sharp convergence rates of Prandtl solutions to Blasius profiles in stronger norms.

## Key findings

- Convergence of solutions in $W^{k,p}$ norms with explicit decay rates.
- Introduction of a new nonnegative quantity $ ext{ extOmega}$ for Blasius solutions.
- Development of a weighted Nash-type inequality for the analysis.

## 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 $\| u - \bar{u}\|_{L^\infty_y} \rightarrow 0$ as $x \rightarrow \infty$. 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.