Absolutely Continuous Solutions in Dissipation

• Absolutely continuous solutions are defined by measures possessing a density with respect to Lebesgue measure, crucial for analyzing energy dissipation in evolution equations.
• They serve to quantify anomalous dissipation in systems like the Navier–Stokes and Euler equations, ensuring energy loss persists in vanishing viscosity limits.
• This concept bridges rigorous PDE analysis with turbulence theory, revealing insights into the regularity and non-uniqueness of weak solutions.

An absolutely continuous solution is a central concept in the modern analysis of evolution equations, stochastic processes, control systems, and spectral theory, defined by the property that its associated measure, process, trajectory, or spectrum is absolutely continuous with respect to an underlying reference measure (typically Lebesgue measure). In applied PDE and mathematical physics settings, especially the analysis of fluid flows and turbulence, absolutely continuous solutions are used to characterize properties of dissipation measures arising as weak limits of nonlinear quantities. These concepts play a critical role in quantifying the mechanisms of energy dissipation, regularity of solutions, and the structure of singularities in nonlinear evolution equations, such as the Navier–Stokes and Euler equations.

1. Definition and Mathematical Context

The absolutely continuous part of a measure μ on a measurable space (for example, on space-time or time alone) refers to the component of μ which is absolutely continuous with respect to another reference measure λ (typically Lebesgue measure); that is, μ_ac ≪ λ, and there exists a density f ∈ L¹ such that dμ_ac = f dλ. When discussing evolution PDEs such as the incompressible Navier–Stokes equations,

$$\partial_t v + (v \cdot \nabla) v + \nabla p = \nu \Delta v + F$$

on, e.g., the torus $\mathbb{T}^4$, the dissipation measure μ arises as the weak* limit (in the sense of measures) of the sequence $\nu |\nabla v_\nu|^2$ as viscosity $\nu \to 0$. The time-projected measure μ_T is then defined as the pushforward: $\mu_T = \pi_\# \mu$, where π projects onto time.

An anomalous dissipation measure is said to possess a nontrivial absolutely continuous part if, in the Lebesgue decomposition

$\mu_T = \mu_{T, ac} + \mu_{T, sing}$

the absolutely continuous part $\mu_{T, ac}$ is not identically zero, i.e., it has nonzero total variation (mass).

In this context, anomalous dissipation refers to the non-vanishing of

$\limsup_{\nu \to 0} \nu \int |\nabla v_\nu|^2 > 0,$

meaning that the energy dissipation rate persists in the zero-viscosity (Euler) limit, a phenomenon connected to the zeroth law of turbulence and Kolmogorov's K41 theory.

2. Main Results: Existence of Absolutely Continuous Dissipation Measures

The principal result constructs, for the forced $4d$ (and related $3+\frac{1}{2}$-dimensional) Navier–Stokes equations, smooth solutions $v_\nu$ with time-independent forces for which, as $\nu \to 0$:

• $v_\nu$ converge weakly* in $L^\infty$ to a solution $v_0$ of the forced Euler equations.
• The dissipation measures $\nu |\nabla v_\nu|^2$ converge (up to a subsequence) weakly* in the sense of measures to a limit $\mu \in \mathcal{M}((0,1)\times\mathbb{T}^4)$.
• The time-projected measure $\mu_T = \pi_\# \mu$ admits a Lebesgue decomposition where the absolutely continuous part $\mu_{T, ac}$ has nontrivial total variation, i.e., $\|\mu_{T, ac}\|_{TV} > 0$, and the singular part $\|\mu_{T, sing}\|_{TV}$ can be made arbitrarily small (i.e., for any given $\beta > 0$ one has $\|\mu_{T, sing}\|_{TV} \leq \beta$).

More precisely, by suitable choices of forces and parameters, $\|\mu_{T, sing}\|_{TV} \leq \beta$ and $\|\mu\|_{TV} \geq 1/4$ hold simultaneously.

This construction rigorously exhibits nontrivial, temporally absolutely continuous anomalous dissipation: in the vanishing viscosity limit, most of the energy dissipation is spread (with a density function) over time, rather than being concentrated on a singular set in time.

3. Energy Balance, Duchon–Robert Distribution, and Strong/Weak Convergence

The local energy balance for weak solutions to the Euler equations is captured by

$$\partial_t \left(\frac{1}{2} |v|^2\right) + \nabla \cdot \left( \left( \frac{1}{2} |v|^2 + p \right) v \right) = -\mathcal{D}[v],$$

where $\mathcal{D}[v]$ is the Duchon–Robert distribution encoding anomalous energy dissipation in the absence of viscosity. In the vanishing-viscosity limit, the measure $\mu$ is “close,” up to a controlled $H^{-1}_{t,x}$-error, to $\mathcal{D}[v_0]$ associated with the limiting Euler solution.

A central subtlety is that, unlike in the strongly convergent case (where $\mathcal{D}[v_0]=0$), for convex integration solutions or defect measure constructions, it is possible for $\mathcal{D}[v_0] \neq 0$, and $\mathcal{D}[v_0]$ may only be a distribution rather than a measure. The result in this work shows that—for the constructed vanishing viscosity sequences—the limit dissipation measure has a nontrivial absolutely continuous component in time, which can (up to error) be associated with the distributional flux $\mathcal{D}[v_0]$.

A key quantitative property is that the kinetic energy profile $t \mapsto \int \frac{1}{2} |v_0(x,t)|^2 dx$ is smooth in time, reflecting the regularity of the absolutely continuous part of the dissipation.

4. Phenomenological Implications and Physical Interpretations

The existence of a nontrivial absolutely continuous part in $\mu_T$ supports the phenomenological scenario in turbulence, where energy dissipation per unit time remains positive in the vanishing viscosity limit—the essence of anomalous dissipation and the zeroth law of turbulence (as proposed by Kolmogorov K41). The mathematical finding here rigorously realizes this by exhibiting energy dissipation mechanisms in which dissipation is distributed over time-lebesgue sets, not merely on singular (measure-zero) sets.

Moreover, non-uniqueness phenomena emerge: different (even smooth) sequences of forcings and initial data may generate vanishing viscosity limits that produce weak Euler solutions with differing dissipation profiles. Importantly, the selection mechanism for these weak solutions via vanishing viscosity is not unique, and having a nontrivial absolutely continuous part in the dissipation measure is consistent with experimental observations and with distributional interpretations of turbulent energy flux.

5. Open Problems and Future Directions

A set of open questions is posed in the paper:

• Strong Convergence and Exact Identification of Dissipation: Can one construct vanishing viscosity sequences with strong convergence (in, e.g., $L^2$ or $L^3$ norms) such that the limit of the viscous dissipation exactly coincides (as measures) with the Duchon–Robert distribution $\mathcal{D}[v_0]$? In current examples, even strong limits may yield singular dissipation (e.g., concentrated in time or on sets of small measure).
• Geometry and Fractal Structure of Dissipation: What is the spatial structure or fractal dimension of the support of the anomalous dissipation? Is it possible to construct solutions where the support of $\mathcal{D}[v_0]$ has a prescribed dimension, possibly relating to intermittency models in turbulence theory (such as the β-model)?
• Selection Criteria via Vanishing Viscosity: How can one define canonical or physically motivated selection criteria for weak Euler solutions arising from vanishing viscosity limits, possibly to ensure uniqueness or to match the statistical structure of turbulent flows? Further research is needed to establish under what conditions the anomalous dissipation measure is exactly the Duchon–Robert distribution and to control the "singular" part in the time variable.

6. Methodological Advances and Connections

The results rely on a novel anomalous dissipation mechanism for the advection-diffusion equation with a divergence-free $3d$ autonomous velocity field, as well as on analysis of the so-called $3+\frac{1}{2}$ dimensional incompressible Navier–Stokes equations. This extends prior work on energy balance, dissipation measures, and convex integration, and builds a bridge between partial regularity theory, measure-valued solutions, and the modern theory of turbulence.

The approach yields detailed information about the time regularity of limiting dissipation measures and provides an explicit route for controlling both the absolutely continuous and singular components with respect to time Lebesgue measure. The construction allows, for any prescribed $\beta>0$, for $\mu_{T,sing}$ to have total variation at most $\beta$, thus making the absolutely continuous part dominant.

7. Summary

Absolutely continuous solutions in the sense of dissipation measures for the incompressible Navier–Stokes and Euler equations rigorously establish that, in certain vanishing viscosity limits, the anomalous energy dissipation is realized as a measure whose projection in time contains a nontrivial absolutely continuous component. This is significant both in demonstrating that the physical picture of continuous-in-time energy dissipation in turbulence is mathematically realizable, and in illuminating the roles of measure decomposition, energy cascade, and non-uniqueness in weak solution selection for the Euler equations. These results open significant avenues for further paper on the strong/weak limit interplay, geometric structure of dissipation, and turbulence modeling.