The transition to instability for stable shear flows in inviscid fluids

In this paper, we study the generation of eigenvalues of a stable monotonic shear flow under perturbations in $C^s$ with $s<2$. More precisely, we study the Rayleigh operator $L_{U_{m,\gamma }}=U_{m,\gamma }{\partial }_x-U_{m,\gamma }^{″}{\partial }_x{\Delta }^{-1}$ associated with perturbed shear flow $\left(U_{m,\gamma }\right(y),0)$ in a finite channel $T_{2\pi }\times [-1,1]$ where $U_{m,\gamma }\left(y\right)=U\left(y\right)+m{\gamma }^2\tilde{\Gamma }(y/\gamma )$ with $U\left(y\right)$ being a stable monotonic shear flow and ${\left\{m{\gamma }^2\tilde{\Gamma }\right(y/\gamma \left)\right\}}_{m\ge 0}$ being a family of perturbations parameterized by m. We prove that there exists $m_{⁎}$ such that for $0\le m<m_{⁎}$, the Rayleigh operator has no eigenvalue or embedded eigenvalue, therefore linear inviscid damping holds. Otherwise, instability occurs when $m\ge m_{⁎}$. Moreover, at the nonlinear level, we show that asymptotic instability holds for m near $m_{⁎}$ and growing modes exist for $m>m_{⁎}$ which equivalently leads to instability.