I will discuss two problems in the dynamics of incompressible fluids. The first is on the metastability of Kolmogorov flow, which is a stable shear flow in the inviscid setting. We will study the persistence of this flow for large time scale when there is small viscosity and in a torus. This is based on a joint work with Ming Xu. The second problem is to understand the manifold structure of all steady flows of 2D Euler equation near a given steady flow. Such manifold structure is important to understand the long time behaviors near a steady flow. We will discuss results near Couette flow (with Chongchun Zeng) and more recent work near general steady flows (with Zhifei Zhang).