On Smooth Vortex Catastrophe of Uniqueness for Stationary Flows of an Ideal Fluid

Abstract

It is well known that the steady-state plane-parallel or spatial axisymmetric flow of an ideal incompressible fluid in a finite-length plane channel or pipe that can be decomposed in powers of spatial coordinates (i.e., is an analytical and, hence, exactly computable flow) is uniquely determined by the inflow vorticity. Under the same boundary conditions, an infinite number of uncomputable phantoms, i.e., infinitely smooth, but nonanalytical flows exist if the domain of a unique analytical flow contains a sufficiently intense vortex cell where the maximum principle is violated for the stream function. A scheme for obtaining an uncomputable vortex phantom for the Euler fluid dynamics equations is described in detail below.