论文标题
关于在二维通道中对称流的稳定性
On the stability of symmetric flows in a two-dimensional channel
论文作者
论文摘要
我们考虑对称流的稳定性 渠道（包括Poiseuille流）。 2015年，Grenier，Guo和 Nguyen在特定的 参数空间的区域，确认正式的渐近学结果 从1940年代开始。我们证明这些流在此外是稳定的 参数空间中的区域。更确切地说，我们证明了 Orr-Sommerfeld操作员 $$ {\ MATHCAL B} = \ big（ - \ frac {d^2} {dx^2}+iβ（u+iλ） 在$$上定义 d（{\ Mathcal B}）= \ {u \ in H^4（0,1）\ ,, \，U^\ prime（0）= u^{（3）}（3）}（0）= 0 \ mbox {and} {and} \，U（u（1）= U^\ prime（1）= 0 \}。 $$在$α\ggβ^{ - 1/10} $或$α\llβ^{ - 1/6} $上的半平面$ \reλ\ geq 0 $。
We consider the stability of symmetric flows in a two-dimensional channel (including the Poiseuille flow). In 2015 Grenier, Guo, and Nguyen have established instability of these flows in a particular region of the parameter space, affirming formal asymptotics results from the 1940's. We prove that these flows are stable outside this region in parameter space. More precisely we show that the Orr-Sommerfeld operator $$ {\mathcal B} =\Big(-\frac{d^2}{dx^2}+iβ(U+iλ)\Big)\Big(\frac{d^2}{dx^2}-α^2\Big) -iβU^{\prime\prime}\,, $$ which is defined on $$ D({\mathcal B})=\{u\in H^4(0,1)\,,\, u^\prime(0)=u^{(3)}(0)=0 \mbox{ and }\, u(1)=u^\prime(1)=0\}. $$ is bounded on the half-plane $\Re λ\geq 0$ for $α\gg β^{-1/10}$ or $α\ll β^{-1/6}$.
