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.
Categories: math.AP math.FA