Mathematical Theorems on Turbulence

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.

• The paper rigorously establishes that energy dissipation in turbulent flows emerges from critical regularity thresholds and weak convergence in Navier-Stokes dynamics.
• It demonstrates that anomalous dissipation and multifractality are inherent consequences of singularities, as evidenced by Onsager’s theorem and precise scaling laws.
• The work connects advanced turbulence theory to model problems like Burgers' shocks and passive scalar transport, offering a framework for future research.

Rigorous Theorems in Turbulence: Constraints, Mechanisms, and Mathematical Structures

Introduction

The paper "Mathematical Theorems on Turbulence" (2601.09619) provides a systematic, theorem-driven exposition of mathematical results constraining the behavior of turbulent flows. Rather than advocating physical or phenomenological models, the work distills rigorous connections between observed features (e.g., anomalous energy dissipation, intermittency, and regularity thresholds) and their mathematical underpinnings within incompressible Euler and Navier-Stokes dynamics, as well as representative model problems. The approach is to elucidate which properties are necessary or forbidden for turbulence, given by theorems subject to minimal hypotheses, and to clarify the architectural relationships among energy dissipation, velocity regularity, and multifractal structure.

Foundational Equations: Euler and Navier-Stokes

The incompressible Euler equations are formulated as a Hamiltonian system constrained by divergence-free velocity, with conservation laws emerging via Noether’s theorem. Regular (classical) solutions in Hölder or Sobolev spaces guarantee conservation of quantities such as energy and circulation. However, the development of singularities—now established even from smooth data in three dimensions and with boundaries [E21, EGM19]—implies that turbulent flow generically escapes any regular regime, necessitating the study of weak solutions.

For realistic fluids, the Navier-Stokes equations with viscosity and external forcing serve as the canonical model. Their non-dimensionalization leads to the Reynolds number ($Re$), encapsulating the interplay between inertial and viscous effects. High-$Re$ flows are characterized by scale proliferation and intermittent features, leading to the consideration of the inviscid ($\nu \rightarrow 0$ or $Re \rightarrow \infty$) limit as a formal definition for fully developed turbulence.

Emergence of Weak Solutions and Energy Dissipation

A central insight is the role of weak convergence and compactness in the vanishing viscosity limit. Sub-sequences of smooth Navier-Stokes solutions are precompact in $L^p$ when uniform scaling bounds on structure functions are available, yielding distributional solutions to the Euler equations [CG12, CV18, DE19]. The observed anomalous dissipation—energy loss persisting as viscosity vanishes—requires that gradients grow in magnitude as $\|\nabla u^\nu\|_{L^2} \sim \nu^{-1/2}$ to sustain a finite dissipation rate, violating the classical expectation of energy conservation.

The local dissipation measure, introduced rigorously (Duchon-Robert distribution), is shown to admit nontrivial, positive mass as a Radon measure in turbulent regimes where regularity is insufficient (sub-$\frac{1}{3}$ derivative in $L^3$), and its support, possibly of lower spatial dimension, indexes the locations and mechanisms of singular mixing and energy loss.

Onsager’s Theorem and Critical Regularity

One of the strongest mathematical results affirmed is Onsager's conjecture: weak Euler solutions with more than $\frac{1}{3}$ spatial derivative in $L^3$ globally conserve energy, while solutions with strictly less can dissipate it [Isett]. The equivalence between structure function scaling, Besov regularity, and the vanishing of energy flux is made precise. Recent advances have established energy conservation for solutions in $BV$ intersected with $L^\infty$ even at the critical regularity, sharply distinguishing incompressible vortex sheets (energy-conserving) from compressible shocks (dissipative) [DI24, DIN24].

A still open question is the fate of energy conservation precisely at the critical $L^3$-Besov threshold; numerical evidence hints at possible vanishing dissipation [IDES25], highlighting subtleties in the excluded middle of Onsager’s regime.

Kolmogorov Laws and Anomalous Scaling

Kolmogorov’s 1941 theory (K41), rooted in statistical arguments and dimensional analysis, is re-expressed with mathematical rigor. The $\frac{4}{3}$ and $\frac{4}{5}$ laws are shown to be deterministic local identities relating third-order structure functions to energy flux for weak Euler solutions [E03, N24]. The inertial range, where these laws operate, is bounded both above and below by precise functions of viscosity and dissipation, and recent theorem [DDII25] quantifies their validity at finite scales and in the limit of vanishing dissipation.

Empirical and numerical studies consistently observe departure from monofractal (linear in $p$) scaling, with anomalous exponents $\zeta_p(p)$ indicating strong intermittency and multifractality in turbulent flow statistics.

Intermittency, Dissipation, and Lower-Dimensional Singularities

The work addresses Landau’s objection to universality in K41: experimental dissipation fields are sparse and concentrated on fractal sets of dimension less than the full space, often near $2.87$ in three dimensions [MS87, MS91]. The main theorem [DDII24, DDII25] establishes that for weak solutions with nontrivial singular dissipation, the support of the dissipation measure must have at least full spatial dimension, and if it concentrates on lower-dimensional sets ($\gamma < d+1$), the structure function exponents are strictly sublinear in $p$, enforcing intermittency.

Sharper quantitative bounds relate the dimension and measure of dissipation-supporting sets to the possible scaling exponents, and experimental fit of these relationships approximates measured multifractal exponents to high accuracy.

Model Problems: Burgers and Passive Scalar

Model equations such as Burgers' (one-dimensional shock dynamics) and advection-diffusion for passive scalars generalize many themes of turbulence. For Burgers, shock formation from smooth data and rigorous bounds on self-regularization are derived, showing that only codimension-one singularities (shocks) support the entropy dissipation measure, with structure functions obeying $\zeta_p = 1$ for $p \geq 3$.

Passive scalar transport, notably in the Kraichnan model with stochastic, rough velocity fields, yields anomalous dissipation precisely when the velocity fails to be Lipschitz; the threshold regularity is characterized in terms of the Obukhov-Corrsin limit. Results [DGP25] establish that the scalar field achieves minimal Besov regularity sufficient for dissipation, and the full multifractal spectrum of exponents can be computed asymptotically for Gaussian velocity ensembles.

Anomalous dissipation is linked to spontaneous stochasticity—the non-uniqueness of backward trajectories in rough velocity fields—a phenomenon rigorously explored in the context of Kraichnan flows and extended mathematically in measure-valued dynamical systems.

Particle Motion, Dispersion, and Lagrangian Theory

The behavior of Lagrangian particles in turbulent flow (classically studied by Taylor and Richardson) is analyzed with respect to the regularity and splitting properties of trajectory bundles. For velocity fields rougher than Lipschitz (e.g., $C^{1/3}$), mean-squared displacement of particle pairs bursts as $t^3$ (Richardson’s law), and the velocity along trajectories becomes only Hölder continuous in time. Mathematical bounds confirm explosive separation in the inertial range, and recent rigorous work offers a Lagrangian representation for energy dissipation, revealing a time irreversibility directly associated with anomalous dissipation.

Implications and Future Directions

The theorems delineated in this work rigorously constrain the plausible features of turbulence by limiting possible regularity and dissipation mechanisms. They clarify that anomalous dissipation and multifractal intermittency are mathematically necessary consequences of singular dynamics, bounded by precise regularity and dimensional measures. These results have implications for modeling, simulation, and understanding of both hydrodynamic turbulence and analogous rough dynamical systems. They also motivate future research on sharp intermittency bounds, universality phenomena, and the Lagrangian dynamics of weak flows.

Prospective developments include extension of these mathematical frameworks to boundary-driven turbulence, viscous anomalies in wall-bounded domains, and refinement of intermittency theory to incorporate probabilistic structure (statistical solutions, random fields). The work also sets the stage for a dynamical system and probability-centered analysis of turbulent phenomena, aligning with early visions of Kolmogorov, Onsager, and Feynman.

Conclusion

This paper systematically develops the mathematical underpinnings of turbulence phenomena via rigorous theorems, contrasting theoretical predictions and empirical observations. It achieves authoritative quantification of energy dissipation, regularity thresholds, and intermittency constraints, serving as a foundational reference for further work in fluid dynamics, statistical physics, and the analysis of rough PDEs.