Existence theorem for the flow of an ideal incompressible fluid in two dimensions

Abstract

We shall prove the existence of the flow of an ideal incompressible fluid in two dimensions under certain initial conditions. Suppose that R is an open connected plane region with a sufficiently smooth boundary. u(x, y, t) and v(x, y, t) are velocity components parallel to the x and y axes respectively and in the positive direction, p(x, y, t) is the pressure, and ais the density. The flow of an ideal homogeneous incompressible fluid is said to exist if u(x, y, t), v(x, y, t), p(x, y, t) and =constant are defined in R throughout some time interval, possessing sufficient differentiability and satisfying the conditions: (1) The divergence is zero and the density is constant. (2) The normal component of the velocity at the boundary is zero. (3) The Euler dynamical equations are satisfied