what is the meaning of " weak form of the N-S equation", regards

When the equations are written in differential form certain assumptions are implied concerning the smoothness of allowable solutions. In order to allow for discontinuous solutions, these assumptions have to be weakened by resorting to an integral form of the equations. With the integral form, a wider range of solutions can be supported, including discontinuous solutions.

As it was mentioned by ag, we can not work with the called "strong form" of the problem due to the requirements, regularities, continuity, ... For that reason, the problem is normally reduced writting it in a "weak form". This is not limited to FEM or SEM types of methods and is the case in general. For example, if you are using FDM, or FVM you are using a "weak form", where to find the solution to the original problem consist in finding a set of values (at the discrete points) that are satisfing the discrete formulation of the problem.