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Existence of a weak solution to the Navier-Stokes equations enjoying the energy equality

This paper proves the existence of a weak solution to the Navier-Stokes equations that satisfies the energy equality by taking the limit of solutions to a mollified system, though it does not establish that all Leray-Hopf weak solutions share this property due to the lack of uniqueness results.

Original authors: Paolo Maremonti, Alfonsina Tartaglione

Published 2026-10-27

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Paolo Maremonti, Alfonsina Tartaglione

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Fluids are everywhere, from the air we breathe to the blood flowing through our veins. When scientists try to predict how these fluids move, they rely on a set of rules known as the Navier-Stokes equations. These equations describe how a fluid's speed and pressure change over time and space. For decades, mathematicians have been able to prove that solutions to these equations exist, meaning a mathematical description of the flow can be found. However, a stubborn gap has remained in our understanding. While we know these solutions generally obey a rule about energy conservation—where the total energy of the fluid should stay constant or decrease slightly due to friction—previous mathematical proofs could only guarantee that energy would not increase. They could not prove that the energy remained perfectly balanced, as physics suggests it should. This uncertainty leaves open the possibility that, under certain conditions, the mathematical models might allow energy to vanish mysteriously, a scenario that would imply a breakdown in our understanding of how fluids behave.

A recent study by Paolo Maremonti and Alfonsina Tartaglione addresses this long-standing gap. The researchers have successfully constructed a specific type of solution to the Navier-Stokes equations that strictly obeys the law of energy conservation. In their work, they demonstrate that it is possible to find a fluid flow description where the total energy at almost every moment, plus the energy lost to friction over time, exactly equals the energy the fluid started with. This result is significant because it proves that the "missing energy" problem is not an inevitable feature of the equations themselves, but rather a limitation of the methods previously used to find solutions. The authors achieved this by refining a classic technique that smooths out the rough edges of the equations to make them easier to solve, and then carefully showing that as these smoothed versions become more precise, they converge on a solution that preserves energy perfectly.

The team did not claim to have solved the entire mystery of fluid dynamics, nor did they prove that every possible way of describing a fluid flow must conserve energy. Instead, they showed that at least one valid solution exists that does. This is a crucial distinction because the equations are known to be complex, and it is not yet known if there is only one unique solution for a given starting condition. If there are multiple solutions, some might still lose energy, but the existence of this new, energy-conserving solution proves that the equations are capable of describing a physically perfect flow. The researchers also identified a specific window of time where this perfect behavior is guaranteed to hold, depending on how much energy the fluid starts with. For fluids starting with a small amount of energy, this window is quite large, while for very energetic fluids, the window is smaller, though it still exists.

To reach this conclusion, the authors employed a method that involves creating a sequence of simpler, approximated versions of the fluid problem. Imagine trying to understand the shape of a jagged rock by first looking at a series of smoother, rounder stones that gradually become more like the original. The researchers used a similar approach, starting with smoothed-out versions of the fluid equations that are easier to handle mathematically. They proved that these approximations behave well and that as the smoothing is removed, the solutions settle into a final, stable form. By carefully tracking how the energy behaves in this transition, they were able to show that the final solution does not lose any energy to mathematical artifacts. This process required a new, specialized tool to handle the numbers involved, allowing them to filter out the parts of the calculation that might have suggested energy loss and focus on the parts that preserve it.

The implications of this work extend beyond just checking a box in a mathematical proof. In the real world, the idea that energy could disappear from a fluid without a physical cause is unsettling. If such a thing were true, it would suggest that turbulence—the chaotic, swirling motion seen in storms or fast-moving water—could act as a hidden sink for energy. The new result suggests that while turbulence is complex, it does not necessarily require a violation of energy conservation to exist. The authors note that their findings align with the idea that any apparent loss of energy in previous models was likely a result of the mathematical tools used, rather than a fundamental property of the fluid itself. This provides a firmer foundation for understanding how fluids move, ensuring that our mathematical models can, at least in some cases, reflect the physical reality of energy conservation without compromise.

While the study focuses on the mathematical existence of these solutions, it also touches on the nature of the "weak" solutions that mathematicians use to describe complex flows. These are solutions that might not be perfectly smooth at every single point but still satisfy the overall rules of the system. The researchers showed that even within this broader, less restrictive category of solutions, it is possible to find one that respects the strict laws of energy. This finding does not rule out the possibility that other, more chaotic solutions might behave differently, but it establishes that the equations are robust enough to support a physically consistent description of fluid motion. The work serves as a reminder that even in the most difficult problems of mathematics, there is often a path to clarity if one looks at the problem from the right angle, using the right tools to bridge the gap between approximation and truth.

Technical Summary: Existence of a Weak Solution to the Navier-Stokes Equations Enjoying the Energy Equality

Problem Statement
The paper addresses the classical problem of the Navier-Stokes initial boundary value problem in a domain $\Omega \subseteq \mathbb{R}^3$ (bounded, exterior, half-space, or whole space). The system is given by:
$v_t + v \cdot \nabla v + \nabla \pi = \Delta v, \quad \nabla \cdot v = 0$
with zero boundary conditions and initial datum $v_0 \in J_2(\Omega)$ .

While the existence of Leray-Hopf weak solutions is well-established, these solutions are known to satisfy only an energy inequality:
$\|v(t)\|_2^2 + 2\int_s^t \|\nabla v(\tau)\|_2^2 d\tau \leq \|v(s)\|_2^2$
The validity of the energy equality (where the inequality becomes an equality) for all $t > 0$ remains an open problem in dimensions $n \geq 3$ . The authors investigate whether there exists at least one weak solution that satisfies the energy equality almost everywhere in time, specifically addressing the gap often attributed to potential turbulence or "anomalous dissipation."

Methodology
The authors employ a constructive approach based on Leray's original mollification technique, avoiding the need for uniqueness results (which are not known for weak solutions). The methodology proceeds as follows:

Mollified System: The authors consider a sequence of regularized Navier-Stokes problems (3.1) where the nonlinear term is smoothed via a Friedrichs mollifier $J_m$ . For smooth initial data, this system admits unique, regular solutions $\{v_m\}$ that satisfy the energy equality exactly.
Convergence Analysis: The core of the proof relies on establishing the strong convergence of the sequence of gradients $\{\nabla v_m\}$ ${\nabla v_{m}}$ to $\nabla v$ $\nabla v$ in specific function spaces.
- Lemma 2.3: A specialized convergence theorem is introduced, acting as a variant of the Lebesgue Dominated Convergence Theorem. It utilizes a cutoff function $p(\alpha, r)$ to handle sequences where standard domination conditions might fail, allowing for the passage to the limit in nonlinear terms.
- Lemma 3.3: The authors prove that the sequence $\{v_m\}$ converges strongly to the weak solution $v$ in $L^q(0, T; J^{1,2}(\Omega))$ for $q \in [1, 2)$ . This is a crucial step, as weak convergence alone is insufficient to pass to the limit in the energy balance.
Auxiliary Function Construction: To recover the energy equality, the authors construct a specific auxiliary function $p(\alpha, r)$ (defined in Section 4.2) involving arctangent and logarithmic terms. This function acts as a weight in the energy equation of the approximating sequence. By carefully analyzing the limit as the parameter $\alpha \to 1^-$ , they demonstrate that the "defect" term (the difference between the energy of the limit and the limit of the energies) vanishes.

Key Contributions and Results
The primary result is Theorem 1.1, which establishes:

Existence of a Specific Weak Solution: For any initial data $v_0 \in J_2(\Omega)$ , there exists a weak solution $v$ (in the sense of Leray-Hopf) that satisfies the energy equality:
$\|v(t)\|_2^2 + 2\int_0^t \|\nabla v(\tau)\|_2^2 d\tau = \|v_0\|_2^2$
for all $t \in (0, \infty) \setminus I$ , where $I$ is a set of Lebesgue measure zero contained within the interval $(0, \theta)$ with $\theta \leq c\|v_0\|_2^4$ .
Regularity and Continuity: The solution $v$ is right-continuous in the $L^2$ -norm at all instants $s$ for which the energy inequality (1.3) holds. The set of possible singular instants $I$ is contained within the specific interval $(0, \theta)$ .
Strong Convergence of Gradients: As a consequence of the energy equality holding for this specific solution, the authors prove (Corollary 4.1) that the sequence of gradients $\{\nabla v_m\}$ converges strongly to $\nabla v$ in $L^2(0, T; L^2(\Omega))$ .
Global Energy Balance: The solution satisfies the global energy dissipation identity:
$2\int_0^\infty \|\nabla v(t)\|_2^2 dt = \|v_0\|_2^2$

Significance and Claims
The paper modestly claims that while it does not prove that every Leray-Hopf weak solution satisfies the energy equality (due to the lack of a uniqueness theorem), it proves the existence of at least one such solution constructed via the standard Leray approximation.

The authors highlight the following implications:

Analytic Constraint: The result imposes a further constraint on the set of possible weak solutions, suggesting that the "gap" in the energy inequality (often interpreted as anomalous dissipation or turbulence) may not be an inherent feature of all weak solutions, but rather a property of specific non-unique solutions.
Relation to Onsager's Conjecture: The authors note that their result allows for a reformulation of conjectures regarding "anomalous dissipation" (analogous to Onsager's conjecture for Euler equations). If the energy equality holds almost everywhere, any discrepancy in energy balance must be attributed to the discontinuity of the $L^2$ -norm at singular times rather than a continuous extra dissipation mechanism.
Well-Posedness Context: The result is presented as relevant to recent discussions on the ill-posedness of the Navier-Stokes equations for very weak solutions. The authors suggest that requiring the energy equality (or inequality) acts as a selection criterion that may rule out certain non-unique, pathological solutions.

The proof is noted to be elementary in its reliance on convergence arguments and specific auxiliary functions, extending previous work on the compatibility of $L^2$ initial data with energy equality. The result holds for both bounded and unbounded domains and can be extended to cases with external forces $f \in L^2(0, T; L^2(\Omega))$ .

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