Evolutions Of Physical Quantities Along Particle Trajectories In Two Dimensional Flows And On Water Waves

We discuss two aspects of the two-dimensional incompressible inviscid fluid. First, the evolutions of the major quantities of the Euler system along the trajec-tories are studied. They are velocity and pressure gradient which is the negative of the force. Conclusions on the asymptotic behavior are obtained by deriving the ordinary differential equation governing the evolutions of the quantities a-long particle trajectories from the Euler equations. Second, the evolutions of the height, the slope of the wave and the speed of a particle on the water wave are discussed. All possible evolutions of these quantities along particle trajectories are listed.Many of the investigation on the Euler equations and the water wave prob-lem are on wellposedness and regularity issues. They discuss the existence and uniqueness of solutions of various regularity. There are relatively few investiga-tions on the properties of the solution. This issue for the solutions to the 2D Euler equations and the water wave problem are taken up in this thesis.