Linear Inviscid Damping in the Presence of an Embedding Eigenvalue

In this paper, we investigate the long-time dynamics of the linearized 2-D Euler equations around a hyperbolic tangent flow \((\tanh y,0)\). A key difference compared with previous results is that the linearized operator has an embedding eigenvalue, which has a significant impact on the dynamics of the linearized system. For the first mode, the dynamics consist of there parts: non-decay part related to the eigenspace associated with the embedding eigenvalue, slow decay part due to the resolvent singularity, and fast decay part related to the inviscid damping. For higher modes, the dynamic is similar to the inviscid damping phenomena in the case without embedding eigenvalues.