Rough dependence upon initial data exemplified by explicit solutions and the effect of viscosity

Abstract

In this article, we present some interesting non-steady explicit solutions to the 2D Euler and Navier–Stokes equations. Explicit calculations on the explicit solutions show that Navier–Stokes (and Euler) equations have the novel property of rough dependence upon initial data in contrast to the sensitive dependence upon initial data found in chaos. Through the explicit calculations, we are able to obtain a lower bound on the norm of the Fréchet derivative of the solution operator at the explicit solutions to the Navier–Stokes equations. The lower bound approaches infinity as the Reynolds number approaches infinity. For Euler equations, this lower bound is indeed infinity. The rough dependence property in the inviscid case is closely related to the theorem of Cauchy. The viscous effect on the theorem of Cauchy and the rough dependence property is also studied.