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.

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Blasius 配置文件的 Global-in-x 稳定性

我们将众所周知的自相似 Blasius 剖面 $[\bar{u}, \bar{v}]$ 表征为解决方案 $[u,v]$ 的下游吸引子，用于 2D 固定 Prandtl 系统。在\cite{Serrin} 中建立了$\| u - \bar{u}\|_{L^\infty_y} \rightarrow 0$ 作为 $x \rightarrow \infty$。在 Blasius 附近的局部数据的情况下，我们的结果通过在更强的规范中建立收敛并通过表征衰减率来进一步推动 \cite{Serrin}。我们分析的核心是“除法估计”，反过来又基于引入了一个新量 $\Omega$，对于 Blasius 解，它是全局非负的。再加上能量级联和新的加权纳什型不等式，这些成分产生了 $u - \bar{u}$ 和 $v - \bar{v}$ 的收敛，在 $W^{ k,p}$ 规范。