Global energy identity for 3D weak Navier–Stokes solutions

Establish whether weak (Leray–Hopf) solutions of the three-dimensional incompressible Navier–Stokes equations satisfy the exact global kinetic energy balance with equality (the energy identity), rather than only the global energy inequality, for all positive times and suitable initial data.

Background

The paper discusses the role of the global energy identity in connecting statistical descriptions of turbulent flows to Kolmogorov’s theory. While Leray–Hopf solutions are known to satisfy only the energy inequality, the equality is a key assumption in deriving Kolmogorov’s power laws for self-similar homogeneous statistical solutions.

Demonstrating the exact energy identity for weak solutions in three dimensions would resolve a long-standing issue and strengthen the mathematical foundations underlying statistical approaches to turbulence.

References

Whether identity (1.3) holds in three space dimensions remains one of the major open problems for weak solutions of the Navier-Stokes equations.

— A Principle of Maximum Entropy for the Navier-Stokes Equations  (2402.14240 - Chen et al., 2024) in Section 1.1 (equation (1.3))

The question of whether every Leray-Hopf weak solution of the 3D NSE satisfies the energy equality is one of the fundamental open problems in the mathematical theory of fluid mechanics ().

— Divergence-Free Approximation in Sobolev and Lebesgue Spaces on General Unbounded Domains with Applications to Energy Equality in Fluid Dynamics  (2609.02769 - Khan et al., 2 Sep 2026) in Section 1, subsection “The Navier-Stokes equations”

Based on Theorem~\ref{th:alpha>2}, we conjecture the following for the true Navier--Stokes equations. There exists a weak solution $u\in L_t\infty L_x2\cap L_t2\dot H_x1([0,\infty)\times\mathbb T3)$ of the 3D Navier--Stokes equations with instantaneous blow-up at $t=0$ obeying both \begin{align*} u(t)\rightharpoonup0\quad\text{weakly in }L2(\mathbb T3)\text{ as }t\to0 \end{align*} and \begin{align*} \lim_{t \to 0+} |u(t)|_{L2} =1. \end{align*}

— Cascade mechanisms for Navier-Stokes blow-up  (2609.26790 - Cheskidov et al., 22 Sep 2026) in Conjecture, Section 1, subsection “Inverse energy cascade and instantaneous blow-up”