Global inviscid limit of 2D, stationary Navier-Stokes and stability of Prandtl expansions

Abstract In this work, we establish the convergence of 2D, stationary Navier-Stokes flows with viscosity $\varepsilon> 0$ , $(u^\varepsilon , v^\varepsilon )$ to the classical Prandtl boundary layer, $(\bar {u}_p, \bar {v}_p)$ , posed on the domain $(0, \infty ) \times (0, \infty )$ : $$ \begin{align*} \| u^\varepsilon - \bar{u}_p \|_{L^\infty_y} \lesssim \sqrt{\varepsilon} \langle x \rangle^{- \frac 1 4 + \delta}, \qquad \| v^\varepsilon - \sqrt{\varepsilon} \bar{v}_p \|_{L^\infty_y} \lesssim \sqrt{\varepsilon} \langle x \rangle^{- \frac 1 2}. \end{align*} $$ This validates Prandtl’s boundary layer theory 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 $\varepsilon \rightarrow 0$ 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.