Stability of steady Euler flows

Suppose φ_s (x, y) is a steady solution of the vorticity equation (3.8). Substitution of ψ = φ_s (x, y) into (3.8) leads to (4.1) ∂ ( φ s , Δ φ s ) = 0. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750087/ed7e0a4b-a39f-4793-9a8e-3af4d8cb07c1/content/eq513.tif"/>