Infinite norm of the derivative of the solution operator of Euler equations

TL;DR

This paper proves that the derivative of the solution operator for Euler equations in Sobolev spaces has infinite norm under certain conditions, and discusses implications for turbulence and initial data sensitivity.

Contribution

It establishes the infinite norm of the derivative of the Euler solution operator in specific Sobolev spaces, providing insights into rough dependence on initial data.

Findings

Derivative norm is infinite for solutions in H^n but not in H^{n+1}.

Discusses turbulence and high Reynolds number Navier-Stokes behavior.

Numerical simulations illustrate rough dependence on initial conditions.

Abstract

Through a simple and elegant argument, we prove that the norm of the derivative of the solution operator of Euler equations posed in the Sobolev space $H^n$, along any base solution that is in $H^n$ but not in $H^{n+1}$, is infinite. We also review the counterpart of this result for Navier-Stokes equations at high Reynolds number from the perspective of fully developed turbulence. Finally we present a few examples and numerical simulations to show a more complete picture of the so-called rough dependence upon initial data.