Energy Equality for Weak Solutions

• Energy equality for weak solutions is the concept that weak fluid dynamics solutions conserve kinetic energy under precise integrability and regularity conditions.
• Sufficient criteria involving L^qL^p and Besov space estimates establish sharp thresholds that prevent anomalous energy dissipation.
• Geometric analysis of singular sets and Type-I blowup scenarios provide practical insights for extending energy conservation to diverse, complex flow regimes.

Energy equality for weak solutions is a fundamental property in the analysis of nonlinear evolution equations, especially in fluid dynamics. It asserts that the kinetic energy (or suitable generalizations thereof) of weak solutions respects the same conservation or balance laws as in the smooth case, provided specific integrability or regularity conditions are satisfied. The effective characterization of energy equality for weak solutions is central to understanding anomalous energy dissipation, thresholds for turbulence and regularity, and the admissibility of weak solutions across classical, fractional, compressible, and non-Newtonian regimes.

1. Formulation of Energy Equality for Weak Solutions

For the incompressible Navier–Stokes equations (NSE), the standard energy equality for smooth solutions on $\mathbb{R}^3$ takes the form: $$\frac{1}{2}\|u(t)\|_{L^2}^2 + \int_{s}^t \|\nabla u(\tau)\|_{L^2}^2\, d\tau = \frac{1}{2}\|u(s)\|_{L^2}^2$$ for any $0 \leq s < t$. For Leray–Hopf weak solutions, energy inequality holds by construction, but energy equality may fail due to possible (singular) irregularities. The investigation centers on identifying minimal conditions under which weak solutions satisfy the equality, eliminating any "defect" or anomalous dissipation.

These conditions are often formulated in terms of mixed-space integrability, e.g., $u \in L^q(0,T; L^p(\mathbb{R}^3))$, or in terms of Besov space regularity (e.g., $u \in L^3(0,T; B^{1/3}_{3,\infty}(\mathbb{R}^3))$) related to the Onsager conjecture.

2. Regularity and Integrability Criteria: Space–Time $L^qL^p$ and Besov Conditions

Classical sufficient conditions for energy equality, such as the Lions–Ladyzhenskaya $L^4(0,T; L^4(\mathbb{R}^3))$ criterion, impose joint space–time integrability. The work in "Conditions Implying Energy Equality for Weak Solutions of the Navier–Stokes Equations" (Leslie et al., 2016) establishes a family of sharp mixed conditions:

• For a Leray–Hopf weak solution $u$ with singularity set $S$ of Hausdorff dimension $d<3$, the optimal admissible exponents depend on $d$:
  • If $p \geq 3$ and $q \leq p$, the condition

  $\frac{2(3-d)}{p} + \frac{5-d}{q} \leq 3-d$

  guarantees energy equality. - If $p \geq 3$ and $p < q$, the inequality is strict.

These are scale-invariant for $d=1$ and recover the classical $L^4L^4$ region, but for $d<1$ (typical for suitable weak solutions via Caffarelli–Kohn–Nirenberg theory), the region is strictly larger, "surpassing" the Lions–Ladyzhenskaya result.

Analogous regularity criteria in Besov or weak Lebesgue spaces (as in (Cheskidov et al., 2018)) take the form $$u \in L^{\beta,w}(0,T; B^{2/\beta + 2/p -1}_{p,\infty})$$ with scaling relation $(2/\beta) + (2/p) < 1$, directly reflecting Onsager's sharp regularity threshold for energy conservation.

3. Geometric Singular Sets and Type-I Blowup

The geometry and measure-theoretic properties of the singularity set $S$—the Onsager singular set—are pivotal. The set consists of space-time points where the solution fails critical 1/3-order regularity, and is characterized by its parabolic Hausdorff dimension $d$. For suitable weak solutions, $d \leq 1$ (Caffarelli–Kohn–Nirenberg).

The new regularity criteria exploit singular set smallness: When $d<1$, weak solutions can admit less restrictive space-time integrability while still ensuring energy equality.

In (Leslie et al., 2016), Type-I blowup scenarios are also treated:

• Type-I in time: $\sup_x |u(x,t)| \leq C/\sqrt{T-t}$,
• Type-I in space: $\sup_{0<t<T} |u(x,t)| \leq C/|x|$.

For Type-I in space, energy equality holds unconditionally. For Type-I in time, energy equality is established assuming $d<1$.

Energy measure techniques (Leslie et al., 2017) provide quantitative frameworks for localizing energy defect and connecting $L^qL^p$ integrability with lower bounds on energy concentration dimension; in particular, Type-I in time blowup implies energy equality for 3D NSE at the blowup time.

4. Fractional Dissipation and Extensions

The fractional Laplacian setting ($(-\Delta)^\gamma$, $0<\gamma<1$) modifies both the regularity criteria and energy equality structure. Weak solutions naturally take values in $$L^2(0,T; H^\gamma(\mathbb{R}^3)) \cap L^\infty(0,T; L^2(\mathbb{R}^3))$$.

The local energy equality for fractional NSE includes commutator terms: $$\begin{aligned} \int |u(t)|^2 \phi - \int |u(s)|^2 \phi &= \ldots - 2\nu \int |\Lambda_\gamma u|^2 \phi - 2\nu \int \Lambda_\gamma u \cdot u \Lambda_\gamma\phi \ &- 2\nu \int \Lambda_\gamma u \cdot [\Lambda_\gamma(u\phi) - (\Lambda_\gamma u)\phi - u \Lambda_\gamma\phi] \end{aligned}$$ Controlling the extra commutator term via multiplier and cutoff function estimates allows derivation of sufficient $L^qL^p$ criteria adapted to the fractional context (Leslie et al., 2016), with dimension-dependent and exponent-dependent optimal choices.

5. Methodologies: Cutoff Functions, Commutator Estimates, and Scale-Invariance

Technical methodologies underlying energy equality proofs include:

• Construction of cutoff functions vanishing near the singular set, whose time-scale $\alpha$ can be optimally chosen dependent on the singular set dimension and operator order. Careful time-localization and independence of spatial and temporal cutoffs (see global mollification with independent boundary cutoffs in compressible settings (Chen et al., 2018)).
• Commutator estimates controlling nonlinear interaction terms, critical in fractional or nonlocal dissipative equations. Lemmas such as

$$\|\Lambda_\gamma(u\phi) - (\Lambda_\gamma u)\phi - u\Lambda_\gamma\phi\|_{L^2} \lesssim \|u\|_{L^p} \| \phi \|^{1-\gamma}_{L^{2p/(p-2)}} \| \nabla \phi \|^\gamma_{L^{2p/(p-2)}}$$

allow the separation and vanishing of error terms in the energy balance.

• Optimization of scale-invariant exponents and cutoff parameters to maximize the admissible region for $(p,q)$ corresponding to energy equality.

6. Implications and Broad Applications

The robust sufficient conditions for energy equality advanced in (Leslie et al., 2016) have multiple consequences:

• They extend classical integrability regions, ruling out anomalous energy dissipation in a wider regime of turbulent or singular solutions.
• In fractional and Type-I blowup settings, they resolve previously open questions regarding the persistence of energy conservation up to critical (potentially singular) times.
• The framework is adaptable to compressible, non-Newtonian, and magnetohydrodynamic systems (see extensions involving fractional dissipation and multiplier space criteria in (Feng et al., 7 Oct 2025, Feng et al., 27 Sep 2024)).
• Sharpness heuristics via intermittency and local energy measures (local dimension, concentration dimension) confirm that any relaxation beyond these criteria can admit energy loss.

7. Generalizations and Current Scope

Current research continues to generalize energy equality criteria:

• Intersection and multiplier space techniques (intersection of $L^qL^p$ with embedding conditions, as in (Veiga et al., 2019)) unify and resolve previously anomalously sensitive conditions (e.g., Shinbrot vs. Serrin classes).
• Energy-variational solutions with defect variables and maximal dissipation selection (Lasarzik, 2021) provide a variational underpinning for both existence and uniqueness—ensuring semiflow and concatenation properties for generalized weak solutions.
• Anisotropic and inhomogeneous function spaces, adapted to primitive and geophysical equations (Necasova et al., 2023, Binz et al., 2023), demonstrate flexibility of the approach for flows with intricate structural or boundary features.

Energy equality for weak solutions thus forms a cornerstone for the analysis of nonlinear PDEs in fluid mechanics, connecting spatial-temporal regularity, singularity structure, scale-invariant estimates, and physical conservation laws across diverse mathematical and physical contexts.