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

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.

References in corpus (3)

• Strong illposedness of the incompressible Euler equation in integer $C^m$ spaces
• Superfast amplification and superfast nonlinear saturation of perturbations as the mechanism of turbulence
• Short term unpredictability of high Reynolds number turbulence --- rough dependence on initial data