AI 中文总结
该研究针对二维可压缩Euler方程光滑单调剪切层的长波谱不稳定性，通过长波展开构造解并推导Wronskian展开，证明了不同马赫数下小波长不稳定模的存在性，揭示了马赫数为√2时长波不稳定性的标度转变规律。
AI 中文摘要
我们研究二维可压缩Euler方程在光滑单调剪切层附近的长波谱不稳定性。通过长波展开构造了可压缩Rayleigh方程的衰减半直线解，并推导了匹配Wronskian的二阶展开式。对于每个固定的马赫数$m>0$，我们证明了当波数足够小时不稳定模的存在性。当$m<\sqrt2$时，这一结论对一类剖面成立，且当$α\to0$时$c_i$趋于正常数。在$m=\sqrt{2}$处，剖面$U_s(Y)=\tanh Y$存在不稳定模，满足$c\to0$且$c_i$的量级为$α^{1/3}$。当$m>\sqrt2$时，同一剖面仍保持不稳定，$c\to c_*(m)\in(0,1)$且正的$c_i$量级为$α$。对应的时间增长率量级分别为$α$、$α^{4/3}$和$α^2$，表明长波不稳定性的标度在$m=\sqrt2$处发生转变。在零厚度极限下，超临界不稳定特征值趋近于实轴，这与文献\cite{CS1,CS2}中超声速可压缩涡旋片的稳定性结果一致。
英文摘要
We study the long-wave spectral instability of the two-dimensional compressible Euler equations around smooth monotone shear layers. We construct decaying half-line solutions of the compressible Rayleigh equation through a long-wave expansion and derive a second-order expansion of the matching Wronskian. For every fixed Mach number $m>0$, we prove the existence of unstable modes for sufficiently small wavenumbers. For $m<\sqrt2$, this holds for a class of profiles, with $c_i$ tending to a positive constant as $α\to0$. At $m=\sqrt{2}$, the profile $U_s(Y)=\tanh Y$ admits an unstable mode with $c\to0$ and $c_i$ of order $α^{1/3}$. For $m>\sqrt2$, the same profile remains unstable, with $c\to c_*(m)\in(0,1)$ and $c_i>0$ of order $α$. The corresponding temporal growth rates are of order $α$, $α^{4/3}$ and $α^2$, respectively, showing a change in the long-wave instability scaling at $m=\sqrt2$. In the zero-thickness limit, the supercritical unstable eigenvalue approaches the real axis, consistently with the stability results for supersonic compressible vortex sheets in \cite{CS1,CS2}.
Comments35 pages
