Chongchun Zeng - Water waves linearized at monotonic shear flows

We consider the 2-dim water wave problem -- the free boundary problem of the Euler equation with gravity and possibly surface tension -- of finite depth linearized at a uniformly monotonic shear flow $U(x_2)$. Our main focus is the eigenvalue distribution. We first show that in contrast to finite channel flows, the linearized capillary gravity wave has two unbounded branches of eigenvalues for high wave numbers, while the gravity waves may have one or two such branches. They may bifurcate into unstable eigenvalues through a rather degenerate bifurcation at the end point of the continuous spectra. Next we consider bifurcations from interior embedded eigenvalues. Under certain conditions, we provide a complete picture of the eigenvalue distribution. The inviscid damping of the linearized capillary gravity waves will also be discussed briefly if time permits. This is a joint work with Xiao Liu.