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

Y. Charles Li

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.

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 , along any base solution that is in but not in , 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.

Paper Structure

This paper contains 7 sections, 1 theorem, 53 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Basic formulation
  3. Proof of the Main Theorem
  4. Example 1 - the trivial one
  5. Example 2 - the simple one
  6. Example 3 - periodic boundary condition
  7. The generic case - numerical simulations

Key Result

Theorem \oldthetheorem

If the Euler equations are posed in the Sobolev space $H^n$ and the base solution is in $H^n$ but not in $H^{n+1}$, then the norm of the formal derivative of the solution operator is infinite when $t>0$.

Figures (2)

  • Figure 1: The solid curve is the numerical result of the super fast growth of perturbations where $\Lambda (t) = \| du(t) \|_{H^3}$. The lower fitting dashed curve is $e^{21.2 \sqrt{t}}$. The closest fitting dashed curve is $e^{30 \sqrt{t}}$.
  • Figure 2: The Reynolds number dependence of the amplifications of perturbations for different Reynolds up to the time $t = 0.3 T_0$, where $\Delta$ is the $H^0$ norm of the perturbation, and $T_0$ is the large eddy turnover time which is around $2$. The dashed curve is a fit to $\sim \sqrt{Re}$, and the red (grey) curve is a fit to $\sim Re^{0.38}$.

Theorems & Definitions (2)

  • Theorem \oldthetheorem
  • proof