Taming Chaotic Fluids — How an Energy Method Finally Pins Down Uniqueness for Compressible Euler Equations | AI Trend Blend

Applied Mathematics · Journal of Mathematical Analysis and Applications 561 (2026) · UNICAMP — Universidade Estadual de Campinas, Brazil · 15 min read

Taming Chaotic Fluids — How an Energy Method Finally Pins Down Uniqueness for Compressible Euler Equations

For the first time, a single constructive framework proves that all weak asymptotic solutions of the compressible Euler equations — covering isentropic gases, isothermal gases, and pressureless fluids — converge to the same unique limit whenever a regular solution exists, resolving a longstanding open question in fluid PDE theory.

Compressible Euler Equations Weak Asymptotic Solutions Radon Measures Energy Method Isentropic Gases Pressureless Fluids Fluid Particle Discretization Well-Posedness PDE

Fluid dynamics has a uniqueness problem. You can construct many different mathematical sequences that all claim to approximate the same physical flow — and for decades, there was no guarantee they would all converge to the same answer. Mathilde Colombeau at the Universidade Estadual de Campinas (UNICAMP) now closes that gap for three physically fundamental fluid systems: isentropic gases, isothermal gases, and pressureless fluids. The key is a cleverly constructed energy functional, rooted in the original physics of fluid particles, that forces every approximating sequence to converge to exactly one limit — and that limit matches the classical smooth solution whenever one exists.

The Problem That Needed Solving

The Euler equations of compressible fluid dynamics describe how density, velocity, and pressure evolve together in a fluid with no viscosity. Written on an $n$-dimensional torus $\mathbb{T}^n$, they take the form of a coupled conservation system for density $\rho$ and momentum $\rho\vec{u}$:

Continuity Equation $$\frac{\partial}{\partial t}\rho + \sum_{i=1}^n \frac{\partial}{\partial x_i}(\rho u_i) = 0,$$

Momentum Equations $$\frac{\partial}{\partial t}(\rho u_j) + \sum_{i=1}^n \frac{\partial}{\partial x_i}(\rho u_j u_i) + \frac{\partial}{\partial x_j} p = 0, \quad j = 1,\ldots,n,$$

State Law $$p = K\rho^\gamma, \quad K \geq 0,\quad 1 \leq \gamma \leq 2.$$

When $K > 0$ and $\gamma > 1$, this models isentropic gas flow. Setting $\gamma = 1$ gives isothermal gas. Setting $K = 0$ gives pressureless flow — clouds of particles that interact only through collisions, relevant to models of dust, granular media, and certain cosmological flows.

The problem is brutal: these equations develop shocks. Classical smooth solutions can only be guaranteed to exist for short times before gradients blow up. Beyond that, one is forced to work with generalized solutions — distributions, measures, limits of sequences. But sequences can accumulate at multiple points. Until now, no one had a way to prove those accumulation points are the same for different approximating sequences, for these multi-dimensional systems, when existence was proved by compactness.

This is precisely what the new paper by Mathilde Colombeau proves — for the first time, combining an existence result obtained by compactness with a uniqueness result in the same framework for a system of partial differential equations.

The Core Achievement in Plain Language

Previous papers constructed many different approximating sequences — each claiming to represent the same fluid flow. Nobody could prove they all converge to the same limit. This paper proves they do, provided one regular approximating sequence exists, and that this common limit coincides with the classical smooth solution when one is available.

Weak Asymptotic Solutions — What They Are and Why They Matter

The concept of a weak asymptotic solution was introduced by Danilov, Omel’yanov, and Shelkovich in 2003 as a tool for finding singular solutions — delta shocks, superpositions of shock waves, and other exotic objects that classical PDE theory cannot directly accommodate. The idea is to construct a family of smooth functions $(\rho_\epsilon, \rho_\epsilon \vec{u}_\epsilon)_\epsilon$ indexed by a small parameter $\epsilon > 0$, which approximately satisfies the Euler equations in the following sense: for every smooth test function $\psi$ on $\mathbb{T}^n$,

Weak Asymptotic Condition $$\int \left\{\frac{\partial}{\partial t}\rho_\epsilon + \sum_{i=1}^n \frac{\partial}{\partial x_i}(\rho_\epsilon u_{i,\epsilon})\right\}\psi\, dx \to 0 \quad \text{as } \epsilon \to 0,$$

with a similar condition for each momentum component. The word “approximate” is doing real work here — these are not solutions of the equations at any fixed $\epsilon$, but they become solutions in the distributional sense as $\epsilon$ tends to zero.

Colombeau’s construction of these sequences goes back to physics. The parameter $\epsilon$ represents the size of a small volume — a fluid particle — containing enough molecules that the macroscopic variables density and velocity are practically constant inside it. Before letting $\epsilon$ go to zero, one studies ordinary differential equations on each fluid particle for fixed $\epsilon$, in the well-behaved classical Banach space $\mathcal{C}(\mathbb{T}^n)$. This is the opposite of the usual approach: rather than deriving the PDE first and then trying to solve it, the method stays at the level of ODEs as long as possible, plugging solutions into the PDE only at the very end when taking the limit.

The approach is constructive — not just an existence proof, but a recipe for computing the solutions, comparable with numerical schemes for ODEs. Numerical simulations based on this framework were presented in earlier papers by Colombeau.

Molecular Agitation — The Physical Ingredient That Ensures Independence

Here is where the construction gets subtle. The simplest version of the ODE system — equations (9) and (10) in the paper — produces weak asymptotic solutions, but those solutions depend on arbitrary choices made during the construction. If different researchers use different arbitrary ingredients, they may get different limits, and it becomes impossible to claim a unique answer.

Colombeau’s fix comes from physics: at the interface between two fluid particles, molecules diffuse in both directions. This molecular agitation is modeled by introducing a perturbation $\mu(\epsilon) > 0$ with $\mu(\epsilon) \to +\infty$ and $\epsilon\mu(\epsilon) \to 0$ as $\epsilon \to 0$. The first condition ensures permanent thermodynamic equilibrium — the physical requirement that there is always molecular exchange. The second ensures the system remains inviscid in the limit.

Modified Velocity with Molecular Agitation $$\tilde{u}^\pm_{i,\epsilon}(x,t) := u^\pm_{i,\epsilon}(x,t) + \mu(\epsilon).$$

This modification transforms the ODE system into an equivalent PDE system with a small diffusion term $\epsilon\mu(\epsilon)\Delta$ — a vanishing viscosity approximation with equal viscosity rates in all equations. The resulting system,

Auxiliary Viscous System $$\frac{\partial}{\partial t}\rho_\epsilon + \sum_{i=1}^n \frac{\partial}{\partial x_i}(\rho_\epsilon u_{i,\epsilon}) = \epsilon\mu(\epsilon)\Delta\rho_\epsilon + r_\epsilon,$$

admits the heat kernel, turning the problem into one that can be analyzed with classical tools. Theorem 1B from earlier work (Colombeau 2023) showed that, with appropriate control on how fast $\epsilon\mu(\epsilon)$ tends to zero, the set of all accumulation points of any weak asymptotic solution is independent of every arbitrary ingredient used in the construction — including the choice of approximations for the initial condition.

That was the existence half. This paper delivers the uniqueness half.

Regular Weak Asymptotic Solutions — Defining the Key Hypothesis

The uniqueness theorem requires a hypothesis: the existence of at least one regular weak asymptotic solution. This is not a strong assumption from a physics standpoint, but it needs precise mathematical formulation. Colombeau’s definition is crisp.

Definition 1 — Regular Weak Asymptotic Solution

A weak asymptotic solution $(w_\epsilon)_\epsilon = (\rho_\epsilon, \rho_\epsilon\vec{u}_\epsilon)_\epsilon$ is regular on $[0,T]$ if there exist constants $M > 0$ and $\eta > 0$ such that for all $\epsilon < \eta$, all $t \in [0, T]$, and all spatial directions $j$,

$$\|w_\epsilon(\cdot, t)\|_\infty \leq M, \quad \left\|\frac{\partial}{\partial t}w_\epsilon(\cdot, t)\right\|_\infty \leq M, \quad \left\|\frac{\partial}{\partial x_j}w_\epsilon(\cdot, t)\right\|_\infty \leq M.$$

These bounds must hold uniformly in $\epsilon$ for all small $\epsilon$.

Three uniform $L^\infty$ bounds: on the solution itself, on its time derivative, and on each spatial derivative. What makes this elegant is that regularity is defined at the level of the approximating sequence, not the limit — and yet it controls the limit through the energy argument. A sequence satisfying these bounds is, in each time slice, a bounded function with bounded first-order distributional derivatives, meaning the limit lives in the Sobolev space $W^{1,\infty}(\mathbb{T}^n)$. By the Rellich–Kondrachov theorem, it is continuous on $\mathbb{T}^n$ — a Lipschitz function in one dimension.

The Uniqueness Theorem — Convergence of Every Sequence to the Same Limit

The main result of the paper is Theorem 1. Its statement is worth reading carefully.

Theorem 1 — Uniqueness and Well-Posedness of Regular Radon Measure Limits

Let $w_0$ be an initial condition for which there exists a regular weak asymptotic solution $(w_\epsilon)_\epsilon$. Then every weak asymptotic solution constructed for the initial condition $w_0$ — whether regular or not — converges to the same limit $w$. This limit $w$ is unique among all Radon measure limits produced by the construction. The solution is well-posed in the $L^1$ sense: small changes in the initial condition produce small changes in the solution.

The proof strategy is elegant in its logic. First, show that the regular weak asymptotic solution itself converges to a limit — this is done by an energy argument showing it forms a Cauchy sequence. Then invoke the independence theorem from earlier work: since all weak asymptotic solutions for the same initial condition share the same set of accumulation points, and since one of them (the regular one) converges to a unique limit, all the others must converge to that same limit too.

The energy argument is the hard analytical core of the paper. Let us trace through how it works.

The Energy Argument — Where the Mathematics Lives

The Change of Variable That Makes Everything Work

The first move is a change of variable that seems almost too simple: replace the density $\rho_\epsilon$ by its reciprocal $\varrho_\epsilon = 1/\rho_\epsilon$. This is permissible because the absence-of-void-region hypothesis guarantees $\rho_\epsilon > 0$ everywhere, and the weak asymptotic solutions are smooth for fixed $\epsilon$. Under this substitution, the viscous system transforms into a form far more amenable to energy estimates. In one spatial dimension, the system becomes

Transformed 1D System $$\frac{\partial}{\partial t}\varrho_\epsilon + u_\epsilon \frac{\partial}{\partial x}\varrho_\epsilon – \varrho_\epsilon \frac{\partial}{\partial x}u_\epsilon = -\epsilon\mu(\epsilon)\varrho_\epsilon^2 \Delta\rho_\epsilon – \varrho_\epsilon^2 r_\epsilon,$$ $$\frac{\partial}{\partial t}u_\epsilon + u_\epsilon \frac{\partial}{\partial x}u_\epsilon – K\gamma \frac{\partial}{\partial x}\frac{\varrho_\epsilon}{\rho_\epsilon^\gamma} = \epsilon\mu(\epsilon)\varrho_\epsilon\Delta(\rho_\epsilon u_\epsilon) – \cdots$$

The key structural feature is the symmetric coupling between the two equations: the off-diagonal terms involve $-\varrho_\epsilon$ and $-K\gamma/\rho_\epsilon^\gamma$ in matching positions. This symmetry is what makes the energy argument close.

The Energy Functional and Problematic Terms

Given two weak asymptotic solutions $(\varrho_{\epsilon_1}, u_{\epsilon_1})$ and $(\varrho_{\epsilon_2}, u_{\epsilon_2})$, the paper introduces an energy functional

Energy Functional $$S(t, \epsilon_1, \epsilon_2) = \int_{\mathbb{T}} \left[\lambda^2(\varrho_{\epsilon_1} – \varrho_{\epsilon_2})^2 + \tau^2(u_{\epsilon_1} – u_{\epsilon_2})^2\right] dx,$$

where $\lambda = \lambda(x,t,\epsilon_1,\epsilon_2)$ and $\tau = \tau(x,t,\epsilon_1,\epsilon_2)$ are weight functions, bounded uniformly in $\epsilon_1, \epsilon_2$ along with their first-order partial derivatives. The goal is to show $S(t,\epsilon_1,\epsilon_2) \to 0$ as both $\epsilon_1$ and $\epsilon_2$ tend to zero.

Differentiating $S$ in time and substituting the equations for the differences $\varrho_{\epsilon_1} – \varrho_{\epsilon_2}$ and $u_{\epsilon_1} – u_{\epsilon_2}$ produces a long expression. Most terms are manageable — they can be bounded by $\text{const} \cdot S(t,\epsilon_1,\epsilon_2)$ directly, using the uniform bounds from condition (26). Two terms, however, resist this treatment.

These are the problematic terms:

Problematic Term 1 (from density equation)

$\text{PT}_1 = -2\int \lambda^2 \varrho_{\epsilon_1}(\varrho_{\epsilon_1} – \varrho_{\epsilon_2})\frac{\partial}{\partial x}(u_{\epsilon_1} – u_{\epsilon_2})\,dx$. Contains a spatial derivative of one difference multiplied by another difference — the product cannot be bounded by $S$ directly.

Problematic Term 2 (from velocity equation)

$\text{PT}_2 = -2\int K\gamma\tau^2 \frac{1}{\rho_{\epsilon_1}^\gamma}\frac{\partial}{\partial x}(\varrho_{\epsilon_1} – \varrho_{\epsilon_2})(u_{\epsilon_1} – u_{\epsilon_2})\,dx$. Contains a spatial derivative of the density difference multiplied by the velocity difference — the same structural obstruction.

The resolution is beautiful. If the weights $\lambda$ and $\tau$ satisfy the balancing condition

Cancellation Condition $$\lambda^2 \varrho_{\epsilon_1} = \tau^2 \frac{K\gamma}{\rho_{\epsilon_1}^\gamma},$$

then $\text{PT}_1 + \text{PT}_2$ combines into a total spatial derivative $\frac{\partial}{\partial x}\left[(\varrho_{\epsilon_1} – \varrho_{\epsilon_2})(u_{\epsilon_1} – u_{\epsilon_2})\right]$, which integrates to zero over the torus. The problematic terms annihilate each other.

The specific choice that achieves this is $\tau = 1$ and $\lambda = \sqrt{K\gamma/\rho_{\epsilon_1}^{\gamma+1}}$, which is bounded above and away from zero thanks to the regularity and no-void hypotheses.

The Cauchy Sequence Conclusion

With the problematic terms eliminated, what remains is a Gronwall-type inequality:

Gronwall Inequality for S $$\left|\frac{\partial}{\partial t}S(t,\epsilon_1,\epsilon_2)\right| \leq \text{const}\cdot S(t,\epsilon_1,\epsilon_2) + o(\epsilon_1,\epsilon_2),$$

where the constant is uniform in $\epsilon_1, \epsilon_2$ and locally uniform in $t$, and where $o(\epsilon_1,\epsilon_2) \to 0$ as both $\epsilon_i \to 0$. Gronwall’s lemma then gives an explicit bound on $S(t,\epsilon_1,\epsilon_2)$ in terms of $S(0,\epsilon_1,\epsilon_2)$ and the $o$-term. Since the initial data for the approximating sequences converge in $L^1$ — and by regularity, therefore also in $L^2$ — the initial value $S(0,\epsilon_1,\epsilon_2)$ tends to zero as well. The sequence $(\varrho_\epsilon, u_\epsilon)_\epsilon$ is thus a Cauchy sequence in $\mathcal{C}([0,T], L^2(\mathbb{T}))^2$.

“As far as we know, it is the first time that one obtains a uniqueness result for weak asymptotic solutions with an existence result obtained by compactness on the other hand for a system of partial differential equations.” — Mathilde Colombeau · J. Math. Anal. Appl. 561 (2026)

Higher Dimensions — Why the Proof Scales Cleanly

One of the more satisfying aspects of the paper is how it handles the extension from one spatial dimension to $n$ dimensions. In 2D, the transformed system takes a matrix form

Matrix Form of 2D System $$\frac{\partial}{\partial t}\mathbf{s} + X\frac{\partial}{\partial x}\mathbf{s} + Y\frac{\partial}{\partial y}\mathbf{s} = 0, \quad \mathbf{s} = (\varrho_\epsilon, u_\epsilon, v_\epsilon)^T,$$

where the matrices $X$ and $Y$ each have all-diagonal entries except for two off-diagonal terms — $-\varrho_\epsilon$ and $-K\gamma/\rho_\epsilon^\gamma$ — placed in symmetric positions. The crucial observation is that there are no mixed spatial derivatives in the system: the $x$- and $y$-directions decouple in the problematic terms. Each direction generates its own pair of problematic terms, and in each direction the same cancellation mechanism works independently.

The structure of the matrices guarantees that this is not an accident: the symmetric placement of the off-diagonal entries is precisely what ensures the cancellation condition can be satisfied simultaneously in every spatial direction. In $n$ dimensions, the matrices are $(n+1) \times (n+1)$ with the same symmetric structure, and the same argument applies with the energy functional extended to

nD Energy Functional $$S(t,\epsilon_1,\epsilon_2) = \int_{\mathbb{T}^n}\left[\lambda^2(\varrho_{\epsilon_1} – \varrho_{\epsilon_2})^2 + \sum_{i=1}^n \tau_i^2(u_{i,\epsilon_1} – u_{i,\epsilon_2})^2\right]dx.$$

The weight choices $\lambda = 1$ and $\tau_i = \sqrt{\rho_{\epsilon_1}^{\gamma+1}/(K\gamma)}$ satisfy the cancellation condition in every direction simultaneously. The nD proof is, in the author’s words, “exactly the same as the 1D proof reproduced in each direction” — and that economy of argument is a mark of a well-designed theory.

Pressureless Fluids — A Cleaner but Different Argument

Pressureless fluids ($K = 0$) require a separate treatment because the symmetric structure of the matrices breaks down. With $K = 0$, there are no off-diagonal entries in the matrices — and therefore no problematic terms to worry about. The velocity equations (67) and (68) in 2D decouple from the density equation and from each other; they form a symmetric system on their own.

This simplicity is exploited directly. Choosing $\lambda = \tau = 1$ immediately gives $\|u_{\epsilon_1} – u_{\epsilon_2}\|_2$ and $\|v_{\epsilon_1} – v_{\epsilon_2}\|_2 \to 0$ from a Gronwall argument on the velocity differences alone. Once the velocity convergence is established, the density equation becomes a linear transport equation driven by a velocity that is already known to converge — and the density Cauchy sequence estimate follows as a consequence.

The proof for pressureless fluids is both shorter and conceptually cleaner than the gas cases. This is physically natural: without pressure, the fluid particles simply drift; without pressure gradients coupling density to velocity, the equations are less entangled.

Coherence With Classical Solutions — Theorem 2

Once uniqueness of the Radon measure limits is established, a natural follow-up question is: if a smooth classical solution exists, does it agree with this limit? The answer, given by Theorem 2, is yes.

Theorem 2 — Coherence With Classical Solutions

Suppose there exists a classical solution with components of class $\mathcal{C}^1$ on $[0,T[\times\mathbb{T}^n$, and a regular weak asymptotic solution, for the same initial condition $w_0$. Then the classical solution coincides with the limit of all weak asymptotic solutions. In particular, the classical solution is unique and equals the regular Radon measure limit.

The proof follows exactly the same energy method as Theorem 1, but instead of comparing two weak asymptotic solutions $(\epsilon_1, \epsilon_2)$, one compares a weak asymptotic solution against the classical solution. The classical solution enters as the “$\epsilon = 0$” term: its PDE residuals vanish identically (no $\mu$, no $r_\epsilon$, no $R_\epsilon$ terms), so the right-hand side of the energy inequality consists only of terms involving $\epsilon\mu(\epsilon) \to 0$ and the remainder terms $r_\epsilon, R_\epsilon \to 0$. Everything on the right side goes to zero, and the argument closes.

This coherence result is not merely a consistency check. It means the framework is conservative: it never contradicts classical theory, and it reduces to classical theory whenever classical theory applies. The weak asymptotic solutions are genuine generalizations — they handle situations where smooth solutions break down, without abandoning compatibility with the smooth case.

What This Paper Does Not Yet Resolve

The uniqueness result depends on the existence of a regular weak asymptotic solution. The paper does not characterize which initial conditions guarantee such a regular solution — that question is left open. In typical PDE theory, regular solutions exist locally in time for smooth initial data, and the smoothness persists as long as gradients remain bounded. But for shock-forming initial data, regularity will eventually fail, and the theorem says nothing about what happens then.

The analysis is also restricted to the $n$-dimensional torus $\mathbb{T}^n$, which avoids boundary effects but is not the setting most relevant for, say, shock waves on the real line. Extensions to unbounded domains — or to initial data that generate delta shocks — would require additional compactness arguments that are not straightforward.

There is also the question of what happens when no regular weak asymptotic solution exists. The remarkable 2005 result of Bianchini and Bressan — referenced in the paper — establishes uniqueness of vanishing viscosity solutions in 1D on the real line for general hyperbolic systems, but its techniques are entirely different and do not extend to the multi-dimensional setting. The gap between 1D and nD remains wide, and this paper does not claim to close it — only to offer a new approach that works cleanly in the regular case, in any dimension.

Honest Scope of the Result

The paper proves uniqueness conditional on regularity — a physically natural condition that holds before shocks form. It does not address uniqueness for general irregular solutions, does not extend to unbounded domains, and does not give criteria for when regularity breaks down. These are genuine open problems that the result does not yet touch.

Broader Significance — Why Physicists and Analysts Should Both Care

The approach of Colombeau is unusual in the PDE literature because it is simultaneously constructive and analytical. The same discretization that generates the numerical schemes also generates the analytic framework for the uniqueness proof. This is not common: typically, existence proofs and numerical analysis proceed on separate tracks, using different tools, with the gap between them papered over by ad hoc arguments. Here, the physical picture — fluid particles of size $\epsilon$, molecules diffusing at their interfaces, velocities defined on a fine grid — feeds directly into the mathematical structure of the energy functional.

The result also has a certain philosophical weight. Weak solutions of hyperbolic conservation laws are notoriously non-unique in the absence of additional conditions (entropy conditions, viscosity criteria, kinetic conditions, etc.). The fact that uniqueness can be recovered — in the compressible Euler setting, in multiple dimensions, for three different fluid models — by appealing to the regularity of one representative solution is a strong structural statement. It says that the irregular solutions in this framework do not scatter into chaos: they are always controlled by the behavior of the smoothest member of their class.

For applications, this matters. Numerical schemes for compressible flow routinely produce multiple candidate solutions, and practitioners need confidence that refining the mesh converges to a single answer. The present paper provides that confidence — in the mathematical setting of weak asymptotic solutions on the torus — for the full range of compressible Euler models from pressureless dust to diatomic gases.

Conclusion — A Proof That Physics and Mathematics Agree

Mathilde Colombeau’s paper solves a problem that had been open since the first papers in this series appeared in 2015: given that weak asymptotic solutions exist — proved by compactness — do they all converge to the same limit? The answer, under the natural condition that a regular solution exists, is yes. The tools are an energy method inspired by Friedrichs’ symmetric hyperbolic theory, a change of variable that exposes a hidden symmetry in the Euler system, and a physical argument about molecular agitation that turns an arbitrary-looking construction into a robust, parameter-independent one.

The result matters beyond its technical scope because of what it says about the relationship between physics and mathematics in fluid dynamics. The Euler equations were derived from physical principles about fluid particles. The weak asymptotic solutions in this paper were constructed by staying close to those same physical principles — discrete fluid particles, molecular agitation, the limit of vanishing viscosity. The fact that this physically grounded construction produces unique, well-posed solutions is not just reassuring; it suggests the physical picture is guiding the mathematics in the right direction.

Uniqueness problems in PDE are usually approached from the top down — writing down an equation and asking what properties its solutions must have. This paper takes a different route: build up from physical principles, extract mathematical structure, and let uniqueness emerge from the coherence of the construction. That the result is clean, dimension-independent, and applicable to three fundamentally different fluid models in a single framework is a genuine contribution to the long and difficult problem of understanding compressible fluid flow beyond the age of smooth solutions.

Read the Full Paper

Published in the Journal of Mathematical Analysis and Applications, Volume 561 (2026). Full proofs, the 1D and nD energy estimates, and the coherence result for all three fluid systems are available via the journal’s website.

📄 Read the Paper (DOI) 🔗 JMAA Journal Page

Academic Citation:
M. Colombeau. Regular weak asymptotic solutions for some systems of compressible Euler equations by an energy method. Journal of Mathematical Analysis and Applications, 561 (2026) 130497. https://doi.org/10.1016/j.jmaa.2026.130497

This article is an independent editorial analysis of a peer-reviewed open-access paper published under the CC BY 4.0 license. Mathematical statements paraphrase the original results; for complete proofs and precise formulations, consult the published paper. The paper was received 7 May 2025 and made available online 10 February 2026.

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