Global-in-$x$ Stability of Steady Prandtl Expansions for 2D Navier-Stokes Flows

In this work, we establish the convergence of 2D, stationary Navier-Stokes flows, $(u^\epsilon, v^\epsilon)$ to the classical Prandtl boundary layer, $(\bar{u}_p, \bar{v}_p)$, posed on the domain $(0, \infty) \times (0, \infty)$: \begin{equation*} \| u^{\epsilon} - \bar{u}_p \|_{L^\infty_y} \lesssim \sqrt{\epsilon} \langle x \rangle^{- \frac 1 4 + \delta}, \qquad \| v^{\epsilon} - \sqrt{\epsilon} \bar{v}_p \|_{L^\infty_y} \lesssim \sqrt{\epsilon} \langle x \rangle^{- \frac 1 2}. \end{equation*} This validates Prandtl's boundary layer theory \textit{globally} in the $x$-variable for a large class of boundary layers, including the entire one parameter family of the classical Blasius profiles, with sharp decay rates. The result demonstrates asymptotic stability in two senses simultaneously: (1) asymptotic as $\epsilon \rightarrow 0$ and (2) asymptotic as $x \rightarrow \infty$. In particular, our result provides the first rigorous confirmation for the Navier-Stokes equations that the boundary layer cannot "separate" in these stable regimes, which is very important for physical and engineering applications.