Developable Surfaces as Generators of the “Isobaric Solutions” to the Euler Equations

Abstract.

The simplest solutions to the Euler equations (1.1) for which the pressure vanishes identically are those representing the motion of lines parallel to a fixed direction $\bar r$ moving in the same direction (each line with an independent, given, constant velocity). Are there many other solutions to this problem? If yes, is there a simple characterization of all the initial data (volume Ω occupied by the fluid at time t = 0 and initial velocity $u_0 (x),x \in \Omega )$ that gives rise to the general solutions? In this paper we show that the answer to both questions is positive. We prove, in particular, that there is a natural correspondence between solutions in R² of this problem and (Cartesian pieces of) developable surfaces in R³. See Theorem 3.