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 $\mathcal{L}_{U_{m,\gamma}}= U_{m,\gamma}\partial_x-U''_{m,\gamma}\partial_x\Delta^{-1}$ associated with perturbed shear flow $(U_{m,\gamma}(y),0)$ in a finite channel $\mathbb{T}_{2\pi}\times [-1,1]$ where $U_{m,\gamma}(y)=U(y)+m\gamma^2\widetilde\Gamma(y/\gamma)$ with $U(y)$ being a stable monotonic shear flow and $\big\{m\gamma^2\widetilde\Gamma(y/\gamma)\big\}_{m\geq 0}$ being a family of perturbations parameterized by $m$. We prove that there exists $m_*$ such that for $0\leq m<m_*$, the Rayleigh operator has no eigenvalue or embedded eigenvalue, therefore linear inviscid damping holds. Otherwise, instability occurs when $m\geq m_*$. Moreover, at the nonlinear level, we show that asymptotic instability holds for $m$ near $m_*$ and a growing mode exists for $m>m_*$ which also leads to instability.