INVISCID DAMPING AND THE ASYMPTOTIC STABILITY OF PLANAR SHEAR FLOWS IN THE 2D EULER EQUATIONS
INVISCID DAMPING AND THE ASYMPTOTIC STABILITY OF PLANAR SHEAR FLOWS IN THE 2D EULER EQUATIONS
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DOI：
10.1007/s10240-015-0070-4
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发表时间：
2015-11-01
影响因子：
6.2
通讯作者：
Masmoudi, Nader
中科院分区：
文献类型：
--
作者：
Bedrossian, Jacob;Masmoudi, Nader
本文证明了T × R上二维无粘Euler方程中剪切流接近平面Couette流的渐近稳定性。也就是说，给定库埃特流的初始扰动在适当的正则类中很小，特别是小于2的Gevrey空间类，速度在L-2中强烈收敛到剪切流，该剪切流也接近库埃特流。涡量通过线性演化渐近地驱动到小尺度，并且当t -> +/-无穷大时弱收敛。由于与弗拉索夫方程中的朗道阻尼的关系，速度场的强收敛有时被称为无粘阻尼。这种收敛是正式推导出在线性水平的开尔文在1887年和它发生在代数率首先计算奥尔在1907年;我们的工作似乎是第一次严格确认这种行为的非线性水平。
We prove asymptotic stability of shear flows close to the planar Couette flow in the 2D inviscid Euler equations on T x R. That is, given an initial perturbation of the Couette flow small in a suitable regularity class, specifically Gevrey space of class smaller than 2, the velocity converges strongly in L-2 to a shear flow which is also close to the Couette flow. The vorticity is asymptotically driven to small scales by a linear evolution and weakly converges as t -> +/-infinity. The strong convergence of the velocity field is sometimes referred to as inviscid damping, due to the relationship with Landau damping in the Vlasov equations. This convergence was formally derived at the linear level by Kelvin in 1887 and it occurs at an algebraic rate first computed by Orr in 1907; our work appears to be the first rigorous confirmation of this behavior on the nonlinear level.