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On "discrete" solutions of the Euler system of gas dynamics

This paper identifies a specific class of initial data for the Euler system of gas dynamics that generates infinitely many "discrete" weak solutions, characterized by finite constant states and increasing entropy profiles that converge to a prescribed terminal state as time approaches infinity.

Original authors: Anna Abbatiello, Eduard Feireisl

Published 2026-08-04

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Anna Abbatiello, Eduard Feireisl

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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

Imagine you are trying to predict the weather. You have a giant, invisible fluid filling the sky—air—that moves, swirls, and crashes into itself. Scientists have a set of rules, called the Euler system, that describe how this fluid should behave. Think of these rules like the ultimate instruction manual for a perfect, frictionless dance. In this dance, the air has no stickiness (viscosity) and no heat loss; it's a pure, idealized flow. For a long time, mathematicians believed that if you knew the starting position of every air molecule, these rules would tell you exactly what happens next, forever. It was like believing that if you drop a ball, physics guarantees it will bounce in one specific, predictable way.

However, there's a catch. In the real world, fluids can get messy. They can form sudden, violent shocks (like a sonic boom) or swirl into chaos. When things get this wild, the "perfect" rules sometimes break down, and the math gets fuzzy. This is where the concept of "weak solutions" comes in. Instead of demanding a perfect, smooth dance for every single molecule, weak solutions allow for a bit of roughness, as long as the overall energy and mass balance out. But here's the twist: recent discoveries have shown that for some starting conditions, the math doesn't just get fuzzy; it gets chaotic in a very specific way. It turns out that for a huge number of starting points, the rules don't just allow for one outcome—they allow for infinitely many different outcomes, all of which seem to follow the laws of physics. It's as if you dropped that ball, and the laws of physics said, "It could bounce left, right, up, down, or turn into a butterfly, and all of those are equally valid."

This paper dives deep into that chaotic playground. The authors, Anna Abbatiello and Eduard Feireisl, investigate a special kind of "wild" solution to the Euler system. They aren't just showing that multiple solutions exist; they are constructing a very specific, almost Lego-like type of solution they call "discrete." Imagine a fluid that doesn't flow smoothly like water in a river, but instead jumps between a finite number of distinct, frozen states, like a pixelated video game character teleporting between a few fixed spots. The paper proves that for a dense set of starting conditions (meaning you can find these conditions almost anywhere you look), you can create an infinite family of these "discrete" solutions. Even more strangely, these solutions can be made to produce more and more entropy (disorder) at will, or they can be forced to settle into a specific, pre-chosen pattern of disorder as time goes on. The authors show that the Euler system is "ill-posed" even when we try to be strict about physics; it allows for these bizarre, non-unique, and highly controllable outcomes.

The Story of the "Pixelated" Fluid

Let's break down what the authors actually found. They are working with the Euler system, which is the mathematical model for a gas (like air) that has no friction and no heat loss. Usually, we expect that if we know the density (how packed the gas is), the momentum (how fast and in what direction it's moving), and the entropy (a measure of disorder or heat), at the start, the future is determined. But the authors show that for a massive collection of starting points, this isn't true.

They discovered a way to build solutions that are "discrete." In the real world, fluids are continuous; they change smoothly. But these solutions are like a digital image made of pixels. The gas doesn't flow smoothly; instead, it exists in a finite number of constant states. Imagine a room filled with gas. In a normal scenario, the gas might swirl gently. In these "discrete" solutions, the gas in one corner is suddenly in State A (a specific density and speed), and in the next corner, it's in State B. It doesn't blend; it just is one thing or another. The authors proved that for a dense set of starting conditions, you can create an infinite number of these pixelated scenarios.

Here is the really mind-bending part: these solutions are not just random. The authors can control them. They showed that you can create a family of these solutions where the amount of disorder (entropy) keeps increasing. It's like having a machine that can generate infinite different versions of a movie, where in one version the characters get slightly more messy, in the next they get even messier, and so on, all while strictly obeying the laws of physics.

Furthermore, they showed that you can force these solutions to end up in a specific "terminal" state. Imagine you want the gas to settle into a very specific pattern of disorder by the time the clock hits a certain time. The authors proved that you can construct a solution that does exactly that, no matter how complex the final pattern is (as long as it's a "Riemann integrable" function, which is a fancy math way of saying it's a pattern you can measure).

Why This Matters (and Why It's Weird)

The authors are essentially saying that the Euler system is "ill-posed." In math-speak, "well-posed" means you have a unique solution that depends nicely on the starting data. "Ill-posed" means the rules are broken. The paper proves that even if you add the rule that "entropy must increase" (which is the Second Law of Thermodynamics, the rule that says things get messier over time), the system still allows for infinitely many different futures.

This is a big deal because it challenges our understanding of how fluids behave. If the math allows for infinite possibilities, how do we know which one is the "real" one? The authors point out that these "discrete" solutions are likely not what happens in the real world. They are "pathological," meaning they are mathematical monsters. They have strange properties, like the gas density staying exactly the same forever while the speed jumps around, or the gas suddenly stopping and starting in a way that feels unnatural.

The paper also touches on the idea of "admissibility." Scientists have tried to create rules to pick out the "real" solution from the infinite fake ones. For example, DiPerna and Dafermos suggested that the "real" solution should be the one that produces the most entropy as fast as possible. The authors show that their "discrete" solutions fail this test. You can always find another solution that produces even more entropy. This suggests that these discrete solutions are indeed unphysical, but their existence proves that the current rules we use to filter them out aren't strong enough to guarantee a single, unique answer.

The Takeaway

So, what's the bottom line? The authors have built a mathematical machine that can generate infinite, pixelated, non-unique solutions to the gas flow equations. These solutions can be tuned to produce more and more disorder or to settle into a specific final pattern. While these solutions are likely not what we see in nature (they are too rigid and weird), their existence proves that the Euler system is fundamentally broken in the sense that it doesn't have a unique answer, even when we try to be very strict about the laws of physics. It's a reminder that in the world of math, sometimes the rules allow for a universe of possibilities, and finding the "real" one requires more than just the basic laws of motion and energy. The paper doesn't say we can't predict the weather; it says that if we rely only on the Euler equations, the math itself says, "I don't know which way the wind will blow, because I can make it blow in a million different ways, and they all look right."

Technical Summary: On "Discrete" Solutions of the Euler System of Gas Dynamics

1. Problem Statement and Context
The paper addresses the issue of well-posedness for the Euler system of gas dynamics, a fundamental model describing the time evolution of compressible, inviscid fluids. While the system is locally well-posed for sufficiently regular initial data, the theory of weak solutions (specifically entropy admissible ones) is known to be ill-posed in the multidimensional setting. Previous work by De Lellis, Székelyhidi, and others utilizing the method of Convex Integration has demonstrated the existence of "wild" initial data—dense in the $L^p$ topology—that generate infinitely many entropy admissible weak solutions.

The specific problem tackled in this work is the characterization of a class of these non-unique solutions that are "discrete." A solution is termed discrete if the state variables (density $\varrho$ , momentum $\mathbf{m}$ , and entropy $s$ ) attain only a finite number of constant values within the space-time domain. The authors aim to extend results previously obtained for the isentropic (barotropic) Euler system by Luo, Xie, and Xin to the full Euler system, while additionally constructing families of solutions with specific entropy production rates and prescribed terminal behaviors.

2. Methodology
The core methodology relies on the Convex Integration technique, adapted to the full Euler system. The approach involves the following steps:

Reformulation: The authors reformulate the problem by fixing the density $\varrho$ and entropy $s$ to be piecewise constant on a finite decomposition of the spatial domain $\Omega = \bigcup_{n=1}^N \Omega_n$ . This reduces the system to finding a momentum field $\mathbf{m}$ that satisfies the momentum balance and a specific kinetic energy constraint on each subdomain $\Omega_n$ .
Reduction to Incompressible-like System: By fixing $\varrho$ and $s$ , the momentum equation is transformed into a system resembling the incompressible Euler equations with a specific pressure term. The constraint on kinetic energy is linked to a constant $\Lambda$ and the pressure $p(\varrho, s)$ .
Application of Discrete Convex Integration: The authors adapt a result by Luo, Xie, and Xin [17] to the transformed problem. This result guarantees the existence of infinitely many weak solutions to the reduced system where the momentum $\mathbf{m}$ takes values from a finite set of constant vectors $\{\mathbf{m}^\ell_n\}$ on each subdomain. These vectors are chosen such that their convex hull contains the origin in a specific phase space, satisfying the necessary subsolution conditions.
Concatenation and Entropy Production: To generate solutions with increasing entropy, the authors divide the time interval $(0, T)$ into segments. On each segment, they apply the discrete solution construction. At the junction points, the momentum is reset to zero (weakly continuous) to allow for concatenation. The entropy increase is achieved not by the momentum vanishing alone, but by selecting a new, higher entropy profile $s_{new} > s_{old}$ for the subsequent segment and adjusting the energy constant $\Lambda$ to satisfy the global energy balance. This process yields a solution where the entropy production rate is positive at the junctions.
Recursive Refinement for Terminal Profiles: To achieve a prescribed terminal entropy profile, the authors employ a recursive refinement of the spatial decomposition as time approaches infinity. By choosing piecewise constant initial entropies on finer and finer partitions that approximate a target Riemann integrable function, they construct a sequence of solutions converging to the desired profile.

3. Key Contributions and Results

The paper establishes the existence of "discrete" solutions for the full Euler system under the following main theorems:

Theorem 2.3 (Discrete Solutions with Increasing Entropy): For a dense set of initial densities and entropies $(\varrho_0, s_0)$ , there exists an initial momentum $\mathbf{m}_0$ such that the Euler system admits infinitely many admissible weak solutions. These solutions are discrete (attaining finite constant states) and possess strictly increasing total entropy profiles. Specifically, for distinct parameters $\lambda_1 < \lambda_2$ , the total entropy of the solution corresponding to $\lambda_2$ is strictly greater than that of $\lambda_1$ .
Theorem 2.6 (Discrete Solutions with Prescribed Initial Kinetic Energy): The result is extended to cases where the initial density $\varrho_0$ and the initial kinetic energy profile $E_{0,kin}$ are prescribed. Infinitely many discrete solutions exist for a dense set of these initial data.
Corollary 2.7 (Discrete Solutions with Prescribed Total Energy): Under specific dimension and adiabatic constant constraints ( $d=2$ or $d=3, \gamma < 5/3$ ), the existence of discrete solutions is shown for prescribed initial density and total energy profiles.
Theorem 2.9 (Discrete Solutions with Prescribed Terminal Entropy): For a monoatomic gas ( $\gamma = 5/3$ ) in 3D, the authors show that for a dense set of initial data, there exists a solution that converges strongly in $L^q$ to a prescribed terminal entropy profile $e_s$ as $t \to T$ . If $e_s$ is piecewise constant, the solution remains discrete for all time.
Long-Time Behavior (Theorems 6.1 and 6.3): The paper demonstrates that the constructed solutions can be made time-periodic or can converge asymptotically to an equilibrium state ( $\mathbf{m}=0$ ) with arbitrary rates of convergence.

4. Significance and Admissibility
The paper highlights that these "discrete" solutions, while mathematically valid entropy admissible weak solutions, exhibit "pathological" properties:

Non-Maximality: The solutions are not maximal with respect to the selection criteria proposed by DiPerna (based on total entropy ordering) or Dafermos (based on the rate of entropy production). The construction allows for the creation of "greater" solutions with higher entropy production rates, implying these discrete solutions are likely unphysical.
Ill-posedness: The results reinforce the ill-posedness of the Euler system in the class of entropy admissible weak solutions, showing that even with the constraint of discrete states, uniqueness fails.
Measure-Valued Solutions: The authors note that convex combinations of these discrete solutions provide examples of measure-valued solutions that also attain only a finite number of constant states.

5. Conclusion
The paper successfully extends the theory of discrete solutions from the isentropic to the full Euler system. By combining convex integration with specific ansatzes for density and entropy, the authors construct a rich family of non-unique solutions. The work underscores that the entropy inequality alone is insufficient to select a unique physical solution, as one can construct solutions that are discrete, satisfy all conservation laws, and yet fail standard maximality criteria for entropy production. The results are presented as a further illustration of the complexity and ill-posedness inherent in the multidimensional Euler equations.

The Story of the "Pixelated" Fluid

Why This Matters (and Why It's Weird)

The Takeaway

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