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

Abstract

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^ε - \bar{u}_p \|_{L^\infty_y} \lesssim \sqrtε \langle x \rangle^{- \frac 1 4 + δ}, \qquad \| v^ε - \sqrtε \bar{v}_p \|_{L^\infty_y} \lesssim \sqrtε \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 $\epsilon \to 0$ and (2) asymptotic as $x\to \infty$. 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.

73 pages. Submitted

References in corpus (6)

• A note on the Prandtl boundary layers
• Optimal Prandtl expansion around concave boundary layer
• Global Steady Prandtl Expansion Over a Moving Boundary
• Boundary layer separation and local behavior for the Steady Prandtl equation
• On the Steady Prandtl Boundary Layer Expansions
• Global $C^{\infty }$ Regularity of the Steady Prandtl Equation

Cited by in corpus (5)

• Boundary Layer Expansions for the Stationary Navier-Stokes Equations
• Remarks on the Steady Prandtl Boundary Layer Expansions
• Global-in-$x$ stability of Prandtl layer expansions for steady magnetohydrodynamics flows over a moving plate
• Validity of Prandtl layer theory for steady magnetohydrodynamics over a moving plate with nonshear outer ideal MHD flows
• Asymptotic Behavior of the Steady Prandtl Equation