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 LUm,gamma = U-m,U-gamma partial derivative(x)- U''(m,gamma)partial derivative(x)Delta(-1) associated with perturbed shear flow (U-m,U-gamma (y), 0) in a finite channel T-2 pi x [-1, 1] where U-m,U-gamma (y) = U(y) + m gamma(2)Gamma(y/gamma) with U(y) being a stable monotonic shear flow and {m gamma 2 Gamma(y/gamma)} (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. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar