We consider the motion of the interface between two superposed fluids with nonzero densities. We assume that the fluids are inviscid, incompressible and irrotational, and initially there is a tangential discrepancy in velocity along the interface. We assume also that the surface tension is zero. We prove the following results: 1. For any initial position (given by a function in Sobolev spaces) of the interface, there is a function in Sobolev space, such that when the initial vortex strength is given by this function, there exists a solution for the above two fluids interface problem for some finite positive time period. 2. For any given pair of initial position of the interface and initial vortex strength (given by functions in Sobolev spaces), if there is a solution of the problem for some positive time period $0\le t\le T$, then the difference of the initial vortex strength and the initial vortex strength determined by the initial position in result 1. is an analytic function. Therefore, the two fluids interface problem generally does not have a solution in Sobolev class beyond initial time unless the initial position and initial velocity satisfy the compatibility conditions given in results 1 and 2.