The well-posedness of two-dimensional ideal flow

The existence, uniqueness, and regularity of solutions to the Euler equations applied to modelling two-dimensional flows is demonstrated. An elementary proof of the existence of regular solutions is presented, and consideration is given to the Euler equation with singular conditions, e.g., the Kelvin-Helmholtz instability featuring an initial velocity discontinuous across a vortex sheet. The Fourier transform of the sheet and the vorticity density are shown to decay exponentially at large wavenumbers by an example of the stability of a stationary solution of the Euler equation, where the velocity is constant in two regions separated by a straight line of discontinuity.