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We consider the 2-dimensional capillary gravity water wave problem – the free boundary problem ofÌýthe Euler equation with gravity and surface tension – of finite depth $x_2 \in (-h,0)$ linearized atÌýa uniformly monotonic shear flow $U(x_2)$. Our main results consist of two aspects, eigenvalueÌýdistribution and inviscid damping. We first prove that in contrast to finite channel flowÌýand gravity wave, the linearized capillary gravity wave has two unbounded branches ofÌýeigenvalues for high wave numbers. Under certain conditions, we provide a complete pictureÌýof the eigenvalue distribution. Assuming there are no singular modes, we obtain the linearÌýinviscid damping. We also identify the leading asymptotic terms of velocity and obtain theÌýstronger decay for the remainders.

This is a joint work with Chongchun Zeng.