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_Um,γ=U_m,γ∂_x−U_m,γ^″∂_xΔ⁻¹ associated with perturbed shear flow (U_m,γ(y),0) in a finite channel T_2π×[−1,1] where U_m,γ(y)=U(y)+mγ²Γ˜(y/γ) with U(y) being a stable monotonic shear flow and {mγ²Γ˜(y/γ)}_m≥0 being a family of perturbations parameterized by m. We prove that there exists m_⁎ such that for 0≤m<m_⁎, the Rayleigh operator has no eigenvalue or embedded eigenvalue, therefore linear inviscid damping holds. Otherwise, instability occurs when m≥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.