Mathematical Theorems on Turbulence

TL;DR

This work assembles rigorous mathematical results that connect turbulence phenomenology to precise regularity and dissipation constraints. By analyzing the Euler and Navier–Stokes equations, Onsager's conjecture, Kolmogorov's laws, and intermittency, it derives both necessary conditions for energy conservation and deterministic manifestations of energy flux in the inertial range, including the $4/5$ and $4/3$ laws. Model problems such as Burgers turbulence and passive scalar advection illuminate the mechanisms of dissipation, intermittency, and Lagrangian trajectories, while Lagrangian dispersion results reveal an arrow of time in 3D turbulence. The framework clarifies how dissipation can persist in the inviscid limit and how intermittency shapes structure-function exponents, offering a principled scaffold for understanding turbulence that guides future probabilistic and dynamical-systems approaches. Overall, the notes provide a rigorous lens on turbulence that links micro-scale regularity to macro-scale energy transfer, while highlighting open questions and the limits of current theory.

Abstract

In these notes, we emphasize Theorems rather than Theories concerning turbulent fluid motion. Such theorems can be viewed as constraints on the theoretical predictions and expectations of some of the greatest scientific minds of the 20th century: Lars Onsager, Andrey Kolmogorov, Lev Landau, Lewis Fry Richardson among others.

Figures (26)

• Figure 1: Flow of water past a cylinder (Album of Fluid Motion VD82).
• Figure 2: Turbulence in periodic box. Second, fourth and sixth order structure functions indicate emergence of non-smooth velocities. LcompDJLW17
• Figure 3: Measurements of second order structure functions and their exponents from numerical simulation D22.
• Figure 4: Dissipation anomaly from experiment S84 and numerics K03
• Figure 5: Lars Onsager and the anomaly O75ES06

Theorems & Definitions (65)

• Theorem 1.1: Local existence in Hölder spaces L25G27
• Theorem 1.2: Finite time singularity in Hölder spaces E21EGM19
• Lemma 2.1
• proof : Proof of Lemma \ref{['Slem']}
• Theorem 2.2: Emergence of weak Euler solutions
• Lemma 2.3: Space regularity implies time regularity
• Proposition 3.1
• proof : Proof of Proposition \ref{['invthm']}
• Lemma 3.2: Pressure Regularity
• proof : Proof of Lemma \ref{['P: pressure CZ']}