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On the Structure of General Weak Solutions of 3D Decaying Turbulence

Min Chul Lee

TL;DR

The paper addresses the temporal structure of general weak solutions to the 3D Navier–Stokes equations on the torus with no forcing, focusing on decaying turbulence. It introduces a simple criterion: a weak solution with monotone-decreasing kinetic energy in time satisfies the strong energy inequality, enabling a comprehensive time-behavior analysis without extra regularity assumptions. The main results establish an equivalence between monotone-energy variants and Leray–Hopf solutions via the differential energy equality, and then develop an order-theoretic framework (partial order, chains, Zorn's lemma) to classify energy evolution and identify potential singular times. This approach yields a structural understanding of energy decay and non-smooth evolution in 3D turbulence, offering a pathway to analyze whether and when singularities may occur in weak solutions.

Abstract

It is shown that a weak solution with monotone-decreasing kinetic energy satisfies the strong energy inequality. Using this criterion, we analyze the behavior with respect to time for all weak solutions without any further assumption on regularity.

On the Structure of General Weak Solutions of 3D Decaying Turbulence

TL;DR

The paper addresses the temporal structure of general weak solutions to the 3D Navier–Stokes equations on the torus with no forcing, focusing on decaying turbulence. It introduces a simple criterion: a weak solution with monotone-decreasing kinetic energy in time satisfies the strong energy inequality, enabling a comprehensive time-behavior analysis without extra regularity assumptions. The main results establish an equivalence between monotone-energy variants and Leray–Hopf solutions via the differential energy equality, and then develop an order-theoretic framework (partial order, chains, Zorn's lemma) to classify energy evolution and identify potential singular times. This approach yields a structural understanding of energy decay and non-smooth evolution in 3D turbulence, offering a pathway to analyze whether and when singularities may occur in weak solutions.

Abstract

It is shown that a weak solution with monotone-decreasing kinetic energy satisfies the strong energy inequality. Using this criterion, we analyze the behavior with respect to time for all weak solutions without any further assumption on regularity.
Paper Structure (6 sections, 3 theorems, 28 equations)

This paper contains 6 sections, 3 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Mathematical Framework and Definitions
  3. Function Spaces
  4. Weak solutions and Their Properties
  5. Main Results
  6. Acknowledgment

Key Result

Lemma 1

Let $u$ be a weak solution to Eq. $(eq:ns-full)$. Then for almost all $t$, where the bracket denotes dual pairing between ${\mathcal{V}}'$ and ${\mathcal{V}}$.

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Proposition 1
  • proof
  • Definition 4
  • ...and 3 more