An Exact Non-stationary Vortex Solution of the Plane Flow and Brownian Motion

Abstract

Based on the proposed conception â€Pseudo-potential†of the incompressible plane flow, an exact non-stationary vortex solution of the Euler’s equation is given, f can be arbitrarily given potential force. It is proved that the given solution can describe the infinitely many unsteady vortices distributed periodically on the whole plane. The Melnikov function method is used to prove Brownian motion appearing along the border region separating different vortices.