论文标题
全球$ x $稳定的prandtl扩展的稳定性2D Navier-Stokes流量
Global-in-$x$ Stability of Steady Prandtl Expansions for 2D Navier-Stokes Flows
论文作者
论文摘要
在这项工作中，我们建立了2D，固定的navier-stokes流量的收敛性，$（u^ε，v^ε）$ to经典的prandtl边界层，$（\ bar {u} _p，\ bar {v} _p）$，在域上构成$（0，\ infty）$（0，\ infty） u^ε-\ bar { v^ε-\sqrtε\ bar {v} _p \ | _ {l^\ infty_y} \ Lessim \sqrtε\ langle x \ rangle^{ - \ frac 1 2}。 \ end {equation*}这验证了Prandtl的边界层理论\ textit {loballyline}在$ x $ - 可变量的一大型边界层中，包括经典blasius概况的整个一个参数家族，并具有清晰的衰减速率。结果同时表明了两种感官的渐近稳定性：（1）渐近为$ε\ rightarrow 0 $和（2）渐近造成的，为$ x \ rightarrow \ rightarrow \ rightarrow \ infty $。特别是，我们的结果为Navier-Stokes方程式提供了第一个严格的确认，即边界层在这些稳定制度中无法“分开”，这对于物理和工程应用非常重要。
In this work, we establish the convergence of 2D, stationary Navier-Stokes flows, $(u^ε, v^ε)$ 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 $ε\rightarrow 0$ 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.
