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 ${\mathcal{L}}_{U_{m,\gamma }}=U_{m,\gamma }{\mathrm{\partial }}_x-U_{m,\gamma }^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\partial }}_x{\mathrm{\Delta }}^{-1}$ associated with perturbed shear flow $(U_{m,\gamma }(y),0)$ in a finite channel ${\mathbb{T}}_{2\pi }\times [-1,1]$ where $U_{m,\gamma }(y)=U(y)+m{\gamma }^2\tilde{\mathrm{\Gamma }}(y\mathrm{/}\gamma )$ with $U(y)$ being a stable monotonic shear flow and $\{m{\gamma }^2\tilde{\mathrm{\Gamma }}(y\mathrm{/}\gamma ){\}}_{m\ge 0}$ being a family of perturbations parameterized by $m$. We prove that there exists $m_{\ast }$ such that for $0\le m<m_{\ast }$, the Rayleigh operator has no eigenvalue or embedded eigenvalue, therefore linear inviscid damping holds. Otherwise, instability occurs when $m\ge m_{\ast }$. Moreover, at the nonlinear level, we show that asymptotic instability holds for $m$ near $m_{\ast }$ and a growing mode exists for $m>m_{\ast }$ which also leads to instability.