Capillary Gravity Water Waves Linearized at Monotone Shear Flows: Eigenvalues and Inviscid Damping

We consider the 2D capillary gravity waves of finite depth x 2 ∈ ( − h , 0 ) x_2 \in (-h, 0) linearized at a monotonic shear flow U ( x 2 ) U(x_2) . The focuses are the eigenvalue distribution and linear inviscid damping. Unlike the Euler equation in a fixed channel where eigenvalues exist only in low wave numbers k k of the horizontal variable x 1 x_1 , we first prove that the linearized capillary gravity wave has two branches of eigenvalues − i k c ± ( k ) -ik c^\pm (k) , where the wave speeds R ∋ c ± ( k ) = O ( | k | ) \mathbb {R}\ni c^\pm (k) = O(\sqrt {|k|}) for | k | ≫ 1 |k|\gg 1 are asymptotic to those of the linear irrotational capillary gravity waves. Under the additional assumption U ≠ 0 U\ne 0 , we obtain the complete continuation of these two branches, which are all the eigenvalues in this (and some other) case(s). In particular, − i k c −