AI 中文总结
该研究针对二维可压缩欧拉方程线性化的单调剪切流，证明密度和无旋速度服从代数增长界、螺线速度经历分量式无粘阻尼，通过两种新能量方法完成证明，恢复了库埃特流的相关速率。
AI 中文摘要
我们研究在$\boldsymbol{\top \times \boldsymbol{\textrm{R}}}$上围绕单调剪切流$\boldsymbol{(U(y),0)}$线性化的二维可压缩欧拉方程。剪切率$\boldsymbol{U'}$严格为正，无需接近任何常数，且变化足够缓慢。对于每个固定马赫数$\boldsymbol{M > 0}$，我们证明密度和无旋速度服从代数增长界，而螺线速度则经历分量式无粘阻尼。尽管非均匀剪切耦合了横向傅里叶频率，排除了库埃特流可用的完全傅里叶约化，但我们仍能恢复无损失的库埃特速率。该证明依赖两个新要素：一是时拟微分能量，它为变系数剪切动力学恢复了强制结构；二是依赖终端时间的高阶与低阶加权能量，它们捕捉剪切混合的长时间效应。
英文摘要
We study 2D compressible Euler equations linearized around monotone shear flows $(U(y),0)$ on $\mathbb{T} \times \mathbb{R}$. The shear rate $U'$ is strictly positive, not necessarily close to any constant, and varies sufficiently slowly. For every fixed Mach number $M > 0$, we prove that the density and the irrotational velocity obey algebraic growth bounds, whereas the solenoidal velocity undergoes componentwise inviscid damping. Although a non-uniform shear couples the transverse Fourier frequencies and precludes the full Fourier reduction available for Couette flow, we are still able to recover the Couette rates without loss. The proof hinges on two new ingredients: a time-dependent pseudodifferential energy that restores a coercive structure for the variable-coefficient shear dynamics, and terminal-time-dependent higher- and lower-order weighted energies that capture the long-time effects of shear mixing.
Comments44 pages
