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Non-conservative $H^{\frac 12-}$ weak solutions of the incompressible 3D Euler equations

Tristan Buckmaster, Nader Masmoudi, Matthew Novack, Vlad Vicol

Abstract

For any positive regularity parameter $β< \frac 12$, we construct non-conservative weak solutions of the 3D incompressible Euler equations which lie in $H^β$ uniformly in time. In particular, we construct solutions which have an $L^2$-based regularity index \emph{strictly larger} than $\frac 13$, thus deviating from the $H^{\frac{1}{3}}$-regularity corresponding to the Kolmogorov-Obhukov $\frac 53$ power spectrum in the inertial range.

Non-conservative $H^{\frac 12-}$ weak solutions of the incompressible 3D Euler equations

Abstract

For any positive regularity parameter , we construct non-conservative weak solutions of the 3D incompressible Euler equations which lie in uniformly in time. In particular, we construct solutions which have an -based regularity index \emph{strictly larger} than , thus deviating from the -regularity corresponding to the Kolmogorov-Obhukov power spectrum in the inertial range.

Paper Structure

This paper contains 94 sections, 62 theorems, 778 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Context and motivation
  3. Ideas and difficulties
  4. Outline of the convex integration scheme
  5. A guide to the parameters
  6. Definitions
  7. Inductive assumptions
  8. Intermittent pipe flows
  9. Lagrangian coordinates, intermittency, and placements
  10. Higher order stresses
  11. Regularity beyond onethird
  12. Specifics of the higher order stresses
  13. Cut-off functions
  14. Velocity and stress cut-offs
  15. Checkerboard cut-offs
  16. ...and 79 more sections

Key Result

Theorem 1.1

Fix $\beta\in (0,1/2)$. For any divergence-free $v_{\mathrm{start}}, v_{\mathrm{end}} \in L^2(\mathbb{T}^3)$ which have the same mean, any $T>0$, and any $\epsilon >0$, there exists a weak solution $v\in C([0,T];H^\beta(\mathbb{T}^3))$ to the 3D Euler equations e:eulereq such that $\left\|v(\cdot,0)

Figures (11)

  • Figure 1: Schematic of the frequency parameters appearing in Definitions \ref{['d:parameters:2.2']} and \ref{['d:parameters:2.4']}.
  • Figure 2: A pipe flow $\mathbb{W}_{q+1,n}$ which is periodized to scale $(\lambda_{q+1} r_{q+1,n})^{-1}=\lambda_{q,n}^{-1}$ is placed in a direction parallel to the $e_2$ axis. Upon taking into account periodic shifts, we note that there are $r_{q+1,n}^{-2}$ many options to place this pipe. This degree of freedom will be used later, see e.g. Figure \ref{['fig:Placing']}.
  • Figure 3: Adding the increment $w_{q+1,0}$ corrects the stress $\mathring R_{q,0}=\mathring R_q$, but produces error terms which live at frequencies that are intermediate between $\lambda_q$ and $\lambda_{q+1}$, due to the intermittency of $w_{q+1,0}$. These new errors are sorted into higher order stresses $\mathring R_{q,n}$ for $1\leq n \leq {n_{\rm max}}$, as depicted above. The heights of the boxes corresponds to the amplitude of the errors that will fall into them, while the frequency support of each box increases from $\lambda_q$ for $\mathring R_{q,0}=\mathring R_q$, to $\lambda_{q+1}$ for $\mathring R_{q+1}$.
  • Figure 4: Adding $w_{q+1,n}$ to correct $\mathring R_{q,n}$ produces error terms which are distributed among the Reynolds stresses $\mathring R_{q,n'}$ for $n+1\leq n'\leq{n_{\rm max}}$.
  • Figure 5: The higher order stress $\mathring R_{q,n}$ is decomposed into components $\mathring R_{q,n,p}$, which increase in frequency and decrease in amplitude as $p$ increases. We use the base of the red boxes to indicate support in frequency, where frequency is increasing from left to right, and the height to indicate amplitudes. Each subcomponent $\mathring R_{q,n,p}$ is corrected by its own corresponding sub-perturbation $w_{q+1,n,p}$, which has a commensurate frequency and amplitude.
  • ...and 6 more figures

Theorems & Definitions (156)

  • Theorem 1.1: Main result
  • Remark 1.2: Corollaries of the proof
  • Definition 2.1: Parameters Introduced in Section \ref{['sec:intro']}
  • Definition 2.2: Parameters Introduced in Section \ref{['ss:inductive:assumptions']}
  • Definition 2.3: Parameters Introduced in Section \ref{['ss:concentratedpipes']}
  • Definition 2.4: Parameters Introduced in Section \ref{['ss:higherorderstresses']}
  • Definition 2.5: Parameters Introduced in Section \ref{['ss:cutoffs']}
  • Remark 3.1: Geometric upper bounds with two bases
  • Remark 3.2: Norms are uniform in time
  • Remark 3.3: Usage of the symbol lesssim and choice of a*
  • ...and 146 more