The global well-posedness of one class of new initial-boundary value problem on incompressible Navier-Stokes equations and the related models in the domain with the boundary is studied. The global existence of a class of weak solution to the initial boundary value problem to two/three-dimensional incompressible Navier-Stokes equation with the pressure-velocity relation at the boundary is obtained, and the global existence and uniqueness of the smooth solution to the corresponding problem in two-dimensional case is also established. Some extends to the corresponding incompressible fluid models such as Boussinesq equation and fluid-structure interaction models etc. are given.