Generalised Energy Conservation (GEC) Condition

• The Generalised Energy Conservation (GEC) Condition is an extension of classical energy laws that permits anisotropic regularity, addressing energy conservation in weak or distributional solutions.
• It utilizes advanced commutator estimates and frequency decomposition techniques to balance regularity between horizontal and vertical velocity components in turbulent flows.
• The anisotropic criteria refine traditional Onsager-critical conditions, offering a flexible framework applicable to both inviscid Euler and viscous Navier–Stokes equations.

Generalised Energy Conservation (GEC) Condition

The Generalised Energy Conservation (GEC) Condition refers to extensions of classical energy conservation laws, inequalities, and admissibility criteria in mathematical physics, particularly in fluid dynamics and gravitation, where standard assumptions—such as isotropic regularity, minimal coupling, or even local conservation—are relaxed or generalized. The principal goal is to determine under what anisotropic or minimal-regularity conditions weak or distributional solutions to the governing equations conserve energy in the absence (or sometimes presence) of viscous, dispersive, or geometric effects. GEC criteria are central to understanding both the mathematical structure of nonlinear PDEs and the onset or suppression of anomalous energy dissipation, especially in turbulent or singular flows.

1. Anisotropic Energy Conservation in Incompressible Fluids

Classical energy conservation criteria, as epitomized by Onsager’s conjecture and its rigorous developments by Cheskidov, Constantin, Friedlander, and Shvydkoy, often require that the velocity field of a 3D Euler solution lies in an isotropic critical Besov space $L^3(0,T; B^{1/3}_{3, c(\mathbb{N})})$. The main advance in (Wang et al., 10 Sep 2025) is the identification of an anisotropic energy conservation class for the 3D incompressible Euler equations, enabled by exploiting the divergence-free condition.

More specifically, write $u = (u_h, u_3)$ with horizontal $u_h$ and vertical $u_3$ components. The core result is:

• $$u_h \in L^3(0,T; B^{\alpha}_{3,c(\mathbb{N})}(\mathbb{R}^3))$$ and $$u_3 \in L^3(0,T; B^{\beta}_{3,\infty}(\mathbb{R}^3))$$ with $1/3 \leq \alpha < 1$ and $\beta \geq (1-\alpha)/2$ suffice for energy conservation:

$$E(t) = \frac{1}{2} \int_{\mathbb{R}^3} |u(x,t)|^2 \, \mathrm{d}x = E(0).$$

This allows the vertical velocity component $u_3$ to belong to the largest critical Besov space $B^{1/3}_{3,\infty}$ while requiring slightly more regularity for $u_h$. The divergence-free constraint (div $u = 0$) ensures, via careful commutator estimates and frequency decomposition (Littlewood–Paley theory), that every nonlinear term in the weak formulations of the energy identity contains at least one horizontal component. This “protects” the estimate from loss of regularity in the vertical direction and provides the technical foundation for the anisotropic GEC class.

2. Comparison with Isotropic Onsager-Critical Criteria

For many years, the prevailing sufficient condition for energy conservation in both the inviscid Euler and viscous Navier–Stokes equations was isotropic in nature:

• $$u \in L^3(0,T; B^{1/3}_{3,c(\mathbb{N})}(\mathbb{R}^3))$$, with $B^{1/3}_{3,c(\mathbb{N})}$ the closure of Schwartz functions under the Besov norm.

The result of (Wang et al., 10 Sep 2025) significantly relaxes this by showing that not all velocity components need meet the full critical regularity. For the viscous (Navier–Stokes) case, a generalization of Lions’s energy conservation class is established: the classical criterion $u \in L^4(0,T; L^4)$ is shown to be a special case of the more general anisotropic relations

• $u_h \in L^{p_1}(0,T; L^{q_1}(\mathbb{R}^3))$ and $u_3 \in L^{p_2}(0,T; L^{q_2}(\mathbb{R}^3))$, with $1/p_1 + 1/p_2 = 1/2$ and $1/q_1 + 1/q_2 = 1/2$.

This structure allows, for example, reduced integrability requirements on one component if the other is sufficiently regular, facilitating a more flexible framework for energy conservation in turbulent and anisotropic flows.

3. Role of the Divergence-Free Constraint and Nonlinear Rearrangement

The divergence-free condition div $u = 0$ is central to the anisotropic GEC result. Its algebraic consequences permit systematic rearrangement of the nonlinear advection term $u \cdot \nabla u$, allowing every term in the nonlinear energy flux to be expressed so as to rely on at least one velocity component with enhanced regularity. For example, the third row of the gradient-multiplied velocity tensor, corresponding to the evolution of $u_3$, leverages the incompressibility to swap certain vertical derivatives for horizontal ones:

$$\left( \begin{array}{ccc} u_1 \partial_1 u_3 & u_2 \partial_1 u_3 & u_3 (\partial_1 u_1 + \partial_2 u_2 ) \end{array} \right),$$

with $$(\partial_1 u_1 + \partial_2 u_2 ) = -\partial_3 u_3$$.

This structural property is the reason the vertical velocity component $u_3$ does not require full critical regularity for energy conservation to hold: the “good” regularity of $u_h$ controls the commutator terms and ensures the decay of energy flux errors.

4. Mathematical Formulation and Commutator Estimates

The proof relies on commutator estimates for Besov spaces, projections onto frequency bands via Littlewood–Paley theory, and careful balancing of regularity between $u_h$ and $u_3$. A basic outline:

• For any test function and mollification parameter, the energy flux defect (difference between mollified products and products of mollified fields) is controlled by terms of the form

$$\| \mathrm{Commutator}[u_h, u_3] \|_{L^1} \leq C \| u_h \|_{L^3_t B^{\alpha}_{3,c(\mathbb{N})}} \| u_3 \|_{L^3_t B^{\beta}_{3,\infty}},$$

and all error terms vanish in the limit as the mollification parameter shrinks, provided the parameters $(\alpha,\beta)$ satisfy $\beta \geq (1-\alpha)/2$ and $\alpha \geq 1/3$.

5. Implications for the Generalised Energy Conservation Condition (GEC)

This anisotropic regularity result broadens the classical GEC condition for incompressible flows and can be viewed as a significant strengthening of Onsager-type criteria. It demonstrates that energy conservation can persist even with directionally localized singularities, provided the interaction of regularity and the PDE’s intrinsic structure (especially incompressibility) is properly harnessed. This has implications for:

• The paper of turbulence, where anisotropic structures (e.g., boundary layers, sheared flows) are ubiquitous.
• Further theoretical generalizations, e.g., to partial regularity or blow-up criteria in more complex fluids or coupled systems.

A potential avenue for future research is to explore how these anisotropic criteria interact with other admissibility criteria (such as entropy conditions in compressible flows) and to clarify whether similar structural decompositions can be exploited in models with less symmetry or additional physical effects.

6. Extension to Viscous Flows and Unified Criteria

For the Navier–Stokes equations, the anisotropic approach unifies and refines the energy conservation classes. For instance, choosing $p_1 = p_2 = q_1 = q_2 = 4$ in the relations above recovers the Lions criterion, and more general exponents allow balance between different spatial directions.

This provides a robust, flexible criterion for energy equality in both theory and numerical analysis, particularly relevant for flows with directional anisotropy or when simulating thin-layer turbulence, channel flows, or other settings where the flow is not statistically isotropic.

Table: Anisotropic GEC Criteria vs. Classical Results

Setting	Regularity (Isotropic)	Regularity (Anisotropic)
Euler	$u \in L^3_t B^{1/3}_{3,c(\mathbb{N})}$	$u_h \in L^3_t B^\alpha_{3,c(\mathbb{N})}$, $u_3 \in L^3_t B^\beta_{3,\infty}$, $\beta \ge (1-\alpha)/2$
Navier–Stokes	$u \in L^4_t L^4_x$	$u_h, u_3$ in anisotropic $L^p L^q_x$ spaces; exponents coupled

The significance of the table is that the anisotropic criteria encompass and extend the isotropic ones, allowing greater regularity flexibility between spatial directions.

7. Context and Broader Impact

The discovery that energy conservation can be established under anisotropic regularity regimes reshapes the understanding of well-posedness and turbulence admissibility in incompressible fluid dynamics. By generalizing Lions’s and Cheskidov–Constantin–Friedlander–Shvydkoy’s results, the approach in (Wang et al., 10 Sep 2025) provides a concrete bridge between mathematical theory and the observed anisotropic structures in experimental and numerical turbulent flows. The methodology utilized here may inspire similar approaches in related nonlinear PDEs where directional effects are prominent, expanding the scope of admissibility criteria beyond isotropic function spaces and opening new questions concerning the role of intrinsic constraints (divergence-free, geometric identities) in energy transport and dissipation.