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Statistical solutions to the Euler system of gas dynamics

This paper proposes a framework for constructing statistical solutions to the compressible Euler system by integrating dissipative measure-valued solutions, a single-step selection procedure based on minimizing Bregman divergence toward maximal entropy equilibrium, and a Markov semigroup construction via push-forward measures.

Original authors: Eduard Feireisl

Published 2026-02-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: 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

The Big Picture: Predicting the Unpredictable

Imagine you have a sealed box filled with a swirling, chaotic gas. You know the rules of physics (how mass, momentum, and energy behave), and you know exactly how the gas starts out. You want to predict exactly where every single molecule will be one hour from now.

In the world of gas dynamics, this is the Euler System. It's the set of equations that describes how a perfect gas moves.

The Problem:
For a long time, mathematicians believed that if you know the starting point, the future is fixed. But recent discoveries showed that for this specific gas system, the math breaks down. If you start with a "messy" (but physically possible) gas cloud, the equations don't just give you one future; they give you infinitely many. It's like throwing a ball and having it land in a million different places at once, all of which seem to obey the laws of physics. This makes the system "ill-posed" (unpredictable).

The Goal:
The author, Eduard Feireisl, wants to fix this. He doesn't want to find just one path for the gas. Instead, he wants to create a Statistical Solution. Think of this not as a single movie of the gas moving, but as a probability map. It tells you: "There is a 70% chance the gas will be here, a 20% chance it will be there, and so on."

The Three-Step Recipe

To build this statistical map, the paper proposes a three-step process.

1. The "Fuzzy" Solution (Dissipative Solutions)

First, the author admits that we can't always find a perfect, sharp picture of the gas. So, he uses a concept called Dissipative Solutions.

The Analogy: Imagine taking a photo of a fast-moving car. If the shutter is too slow, the car looks blurry. That blur isn't a mistake; it contains information about all the places the car could have been during that split second.
In math terms, instead of a single point for the gas, we use a "cloud of possibilities" (a measure) that captures all the potential ways the gas could behave. This allows the math to exist even when the gas gets chaotic.

2. The "Best Guess" Filter (Selection Criterion)

Now we have a cloud of infinite possibilities. We need to pick the one that makes the most physical sense.

The Rule: The Second Law of Thermodynamics says that in a closed system, entropy (a measure of disorder) tends to increase until it hits a maximum. Nature loves to settle into the most disordered, stable state possible.
The Method: The author proposes a "single-step selection." Imagine you have a bag of different possible futures for the gas. You want to pick the one that is closest to the "perfectly settled" state (maximum entropy).
The Tool: He uses something called Bregman Divergence. Think of this as a special ruler that measures the "distance" between a chaotic, messy future and the calm, stable equilibrium. The author's rule is simple: Pick the future that minimizes this distance.
Why it's special: Previous methods required a long, complicated checklist of rules to pick a winner. This method uses just one rule: "Get as close to maximum entropy as possible."

3. The "Time Machine" (Markov Semigroup)

Once we have a rule to pick the best future from the current state, we can build a machine that runs time forward.

The Analogy: Imagine a video game where, at every frame, the computer looks at all possible next moves, picks the one that best fits the "entropy rule," and then moves to that frame.
Mathematically, this creates a Markov Semigroup. It's a system that takes a probability distribution of the gas at time $t=0$ and pushes it forward to time $t=1$ , then $t=2$ , and so on, always applying that "closest to equilibrium" rule. This creates the Statistical Solution.

The Surprising Discovery: The Chaos Fades

The paper proves a very cool result about what happens if you let this system run for a very long time.

The "Energy Defect": When the gas is chaotic (turbulent), there is a gap between the total energy we started with and the energy we can actually see in the smooth, average picture. This gap is called the "energy defect." It's like the energy lost to the "blur" in our photo.
The Result: The author proves that as time goes on ( $t \to \infty$ ), this energy defect vanishes.
The Meaning: Even though the gas starts out chaotic and unpredictable, the "best guess" solution eventually settles down. The blur clears up. The statistical solution becomes a single, clear, predictable path. The turbulence dies out, and the gas behaves like a normal, calm fluid again.

Summary

The Problem: Gas equations are chaotic and have infinite answers.
The Fix: Use "fuzzy" math (dissipative solutions) to handle the chaos.
The Filter: Pick the answer that is closest to the most disordered, stable state (maximum entropy) using a single, simple rule.
The Result: This creates a reliable statistical map of the gas's future.
The Bonus: Over time, the chaos disappears, and the gas settles into a calm, predictable state.

The paper doesn't claim this solves real-world engineering problems immediately, but it provides a solid mathematical foundation for how to think about and predict chaotic gas flows when the standard rules fail.

Technical Summary: Statistical Solutions to the Euler System of Gas Dynamics

Problem Statement
The paper addresses the well-posedness of the Euler system for compressible perfect fluids in dimensions $d=1, 2, 3$ . While the system is locally well-posed for sufficiently regular solutions, it is known that singularities (shock waves) can form in finite time even for smooth initial data. In the class of weak (distributional) solutions, uniqueness fails; specifically, recent results using convex integration demonstrate that for a dense set of initial data in dimensions $d=2, 3$ , there exist infinitely many entropy-admissible weak solutions. This "wild" behavior renders the classical initial value problem ill-posed. To recover a deterministic evolution, the paper proposes a framework of statistical solutions, where the initial data are treated as random variables, and the evolution is described by a Markov semigroup acting on the space of probability measures.

Methodology
The approach extends the class of admissible solutions beyond standard weak solutions to dissipative (measure-valued) solutions, a concept previously introduced to accommodate limits of consistent numerical approximations. These solutions are characterized by a parametrized measure (Young measure) and a defect measure representing oscillations and concentrations.

To select a unique physically relevant trajectory from the non-unique set of dissipative solutions, the author proposes a single-step selection criterion based on the Second Law of Thermodynamics. The methodology proceeds as follows:

Dissipative Solutions: The solution space is defined as the set of dissipative solutions satisfying the continuity, momentum, and energy balance equations in a distributional sense, along with an entropy inequality. These solutions allow for an energy defect (loss of total energy due to oscillations) and satisfy a weak-strong uniqueness property.
Selection Criterion: Instead of using an infinite family of cost functionals (as in previous Krylov-type selections), the paper minimizes a single cost functional. This functional is the time-integral (with an exponential weight $e^{-t}$ $e^{- t}$ ) of the Bregman divergence of the solution state relative to a constant equilibrium state that maximizes entropy for the given total mass and energy.
- The cost functional is defined as $F(\varrho, m, S) = \int_\Omega [E(\varrho, m, S) - \vartheta S] \, dx$ , where $\vartheta$ is the equilibrium temperature derived from the total energy $E_0$ and mass $M_0$ .
- The selection seeks the solution $(\varrho, m, S)$ that minimizes $\int_0^\infty e^{-t} F(\varrho, m, S)(t, \cdot) \, dt$ over the set of all dissipative solutions emanating from the initial data.
Markov Semigroup Construction: Once the unique "admissible dissipative solution" is selected for any given initial data, the mapping from initial data to the solution at time $t$ is shown to be Borel measurable. This allows for the construction of a Markov semigroup on the space of probability measures via the push-forward of the initial measure.

Key Contributions and Results

Existence and Uniqueness of Admissible Solutions: The paper proves that for any initial data in the phase space, there exists a unique admissible dissipative solution selected by the single-step minimization of the Bregman distance. This solution mapping satisfies the semigroup property.
Vanishing Energy Defect: A central result (Proposition 3.2) establishes that for the selected admissible dissipative solution, the energy defect $dE(t) = E_0 - \int_\Omega E(\varrho, m, S)(t, \cdot) \, dx$ $d E (t) = E_{0} - \int_{Ω} E (ϱ, m, S) (t, \cdot) d x$ vanishes as $t \to \infty$ $t \to \infty$ .
- The proof utilizes a contradiction argument: if the energy defect remained positive, one could construct a concatenated solution with a lower cost functional by adjusting the entropy at a specific time, violating the minimality assumption.
- This implies that the selected solution approaches a weak solution (where the energy defect is zero) in the long-time limit.
Statistical Solution Construction: The paper defines a statistical solution as the family of Markov operators acting on probability measures, generated by the push-forward of the initial distribution through the selected solution semigroup.
Properties of Statistical Solutions: The constructed statistical solution possesses:
- The Markov semigroup property ( $M_{t+s} = M_t \circ M_s$ ).
- Regularity (Borel measurability).
- Asymptotic vanishing of the expected energy defect: $\int dE \, dM_t[\sigma] \to 0$ as $t \to \infty$ for any initial measure $\sigma$ .

Significance and Claims
The paper claims that this framework provides a rigorous platform for constructing statistical solutions to the Euler system without relying on arbitrary choices of cost functionals. The selection criterion is motivated by the classical statement of the Second Law of Thermodynamics (Clausius): "The energy of the world is constant; its entropy tends to a maximum." By minimizing the distance to the maximal entropy equilibrium, the method selects a solution that is physically relevant.

The authors note that the decay of the energy defect is consistent with the decay of turbulent phenomena, as the energy defect is directly related to the magnitude of the Reynolds stress in deterministic turbulence models.

Limitations and Open Questions
The paper modestly acknowledges that it does not prove the selected solution is a weak solution for all finite times $t > 0$ . The result that the energy defect vanishes is asymptotic ( $t \to \infty$ ). The authors pose the open question of whether the selected solution is a weak solution for all $t$ . They suggest a partial answer: if the minimizer of a modified functional with a parameter $\lambda$ is independent of $\lambda$ for large $\lambda$ , then the solution is a weak solution. This connection is left for future investigation. Additionally, the paper compares its single-step selection to a previous two-step procedure (maximizing entropy then minimizing energy), noting that if the single-step solution is a weak solution, it would coincide with the result of the first step of the two-step procedure.