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Equivalence of entropy solutions and gradient flows for pressureless 1D Euler systems

This paper establishes the equivalence between Lagrangian and entropy solutions for pressureless 1D Euler systems by linking Ole\u012bnik's E-condition to gradient flow characterizations, thereby enabling the definition of unique post-blow-up solutions and describing their asymptotic behavior for the Euler--Poisson system with quadratic confinement.

Original authors: José Antonio Carrillo, Sondre Tesdal Galtung

Published 2026-06-25

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: José Antonio Carrillo, Sondre Tesdal Galtung

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: Two Different Maps to the Same Destination

Imagine you are trying to predict how a crowd of people moves through a hallway. Some people might bump into each other, stick together, or bounce apart. This paper is about a specific type of crowd: one where the people have no "pressure" (they don't push back against each other like a gas) and they are influenced by forces that pull them together or push them apart.

The authors, José Antonio Carrillo and Sondre Tesdal Galtung, discovered something surprising: Two completely different mathematical methods for predicting this crowd's movement actually give you the exact same answer.

Think of it like two different GPS apps. One app uses a "particle" approach (tracking every single person), and the other uses a "wave" approach (looking at the density of the crowd as a whole). Usually, these two methods might disagree on what happens when a traffic jam forms. But this paper proves that for this specific type of crowd, both apps are perfectly synchronized.

The Two Methods Explained

1. The "Sticky Particle" Method (Lagrangian Solutions)

The Analogy: Imagine you have a line of marbles on a track.

The Setup: Each marble has a mass and a velocity. They are influenced by forces (like magnets pulling them together or repelling them).
The Problem: What happens when two marbles crash? In the real world, they might bounce, or they might stick together like glue.
The Rule: This method uses a "sticky" rule. If two marbles crash, they stick together and move as one heavier marble. If they are repelling each other, they might crash, stick for a moment, and then split apart if the repulsive force gets strong enough.
The Math: The authors treat the movement of these marbles as a "gradient flow." Imagine the marbles are rolling down a hill of energy. They always try to find the path of least resistance that keeps them in order (they can't pass through each other). If they hit a wall (a collision), they slide along the wall rather than breaking the rules.

2. The "Traffic Wave" Method (Entropy Solutions)

The Analogy: Instead of tracking individual marbles, imagine you are looking at a traffic report on a highway. You see a "wave" of cars.

The Setup: You track the density of cars. Where are they packed tight? Where are they spread out?
The Problem: When traffic jams form, you get "shocks" (sudden stops) or "rarefactions" (sudden openings).
The Rule: This method uses "Entropy." In physics, entropy often means disorder, but here it acts as a "quality control" filter. It asks: "Is this traffic jam physically possible?"
- If cars are crashing into each other from behind, that's a valid shock (a traffic jam).
- If cars are suddenly speeding up out of nowhere to create a gap, that's impossible. The "Entropy" rule filters out these impossible scenarios.
The Math: They turn the movement of the crowd into a single equation (a conservation law) that describes how the "wave" of density moves.

The "Aha!" Moment: The Connection

The paper's main breakthrough is proving that the "Sticky Particle" method and the "Traffic Wave" method are actually the same thing.

The Bridge: The authors found a secret handshake between the two methods.
- In the Particle world, when marbles stick together, their velocity is calculated by a specific "projection" (like a shadow cast on a wall).
- In the Wave world, when a traffic jam forms, the speed of the jam is calculated by the "Rankine-Hugoniot condition" (a rule about how much mass and momentum is conserved).
- The paper proves that these two calculations are mathematically identical.
- Furthermore, the rule that keeps particles from passing through each other (the "Normal Cone") is exactly the same as the "Entropy Condition" that keeps traffic waves from behaving impossibly.

Why Does This Matter? (The "Blow-Up" Problem)

In the real world, sometimes these systems break down. Imagine a crowd running so fast that they all crash into a single point at the exact same time. In math, this is called a "blow-up" or a "singularity."

Before this paper: If a classical solution (a smooth, perfect prediction) broke down, mathematicians had to guess what happened next. Did the crowd stick? Did they bounce? Different methods gave different answers.
After this paper: Because the two methods are equivalent, we now have a unique, agreed-upon way to describe what happens after the crash.
- If the force is attractive (like gravity), the crowd sticks together forever.
- If the force is repulsive (like magnets pushing apart), the crowd might crash, stick for a moment, and then split apart again. The paper proves exactly when and how they split.

Real-World Examples from the Paper

The authors tested this equivalence on a few specific scenarios:

The Repulsive Force (Magnets):
- Imagine particles that hate each other. If they crash, they might stick briefly. But because they hate each other, they eventually push apart. The paper shows that the "sticky particle" view and the "traffic wave" view agree on exactly when they split apart. It turns out they split apart the moment the "push" becomes stronger than the "stick," ensuring energy doesn't magically appear out of nowhere.
The Damped System (Friction):
- Imagine the crowd is moving through thick mud (damping). Even if they crash and stick, the friction slows them down. The paper shows that eventually, no matter how chaotic the start, the crowd settles into a calm, uniform distribution. The "sticky" method and the "wave" method both predict this calm state perfectly.

Summary

This paper is like finding out that two different languages (Particle Language and Wave Language) are actually just translations of the same story.

Old View: We had two ways to solve the problem, and we weren't sure if they agreed when things got messy (crashes).
New View: They always agree. The "sticky" rule for particles is the exact same rule as the "entropy" rule for waves.
Result: We can now confidently predict the future of these pressureless crowds, even after they crash, knowing that the prediction is unique and physically sound.

Technical Summary: Equivalence of Entropy Solutions and Gradient Flows for Pressureless 1D Euler Systems

Problem Statement
The paper investigates distributional solutions to the Cauchy problem for the one-dimensional pressureless Euler system:
$\partial_t\rho + \partial_x(\rho v) = 0, \quad \partial_t(\rho v) + \partial_x(\rho v^2) = f[\rho],$
where $\rho$ is a probability measure with bounded second moments and $v$ is a velocity field. The forcing term $f[\rho]$ is generated by the fluid itself, with specific attention given to the Euler–Poisson system (attractive or repulsive) and systems with quadratic confinement and linear damping. A central challenge in this field is the formation of singularities (mass concentration or "blow-up") where classical solutions break down. The paper addresses the need to extend solutions beyond these singularities and to establish a unique, physically relevant continuation.

Two distinct solution frameworks have been developed for such systems:

Lagrangian Solutions: Introduced by Brenier, Gangbo, Savaré, and Westdickenberg [7], these are formulated as $L^2$ -gradient flows in the space of probability measures, utilizing optimal transport maps and differential inclusions involving normal cones.
Entropy Solutions: Studied by Nguyen and Tudorascu [36, 37] for the Euler–Poisson system, these are defined via scalar conservation laws for the cumulative distribution function, relying on the Oleĭnik E-condition for admissibility.

While both frameworks yield unique solutions under specific assumptions, their equivalence had not been rigorously established for the general second-order Euler system, particularly in the presence of mass concentration and repulsive forces where particle trajectories may cross or separate.

Methodology
The authors establish the equivalence between these two concepts by constructing a direct correspondence between the geometric structures governing each framework. The core of the methodology involves:

Variable Transformation:
- For Lagrangian solutions, the primary variable is the optimal transport map $X(t, m)$ , a non-decreasing function pushing the Lebesgue measure to $\rho(t)$ .
- For entropy solutions, the primary variable is the distribution function $M(t, x)$ , the right-continuous inverse of $X$ .
Geometric Correspondence:
The paper demonstrates that the mechanisms selecting unique solutions in both frameworks are mathematically identical:
- Shock Admissibility: The condition that the velocity of the transport map $\dot{X}$ is the $L^2$ -projection of the prescribed velocity $U$ onto the tangent cone of the cone of non-decreasing functions ( $\dot{X} = P_{TXK}U$ ) is shown to be equivalent to the Rankine–Hugoniot condition for shocks in the distribution function $M$ .
- Selection Principle: The requirement that the residual $U - \dot{X}$ lies in the normal cone $N_{X}K$ (ensuring the solution stays within the admissible cone) is proven to be equivalent to the Oleĭnik E-condition for the scalar conservation law governing $M$ . This equivalence relies on a characterization of the normal cone due to Natile and Savaré [35].
Flux Function Derivation:
The authors derive a scalar conservation law $\partial_t M + \partial_x U(t, M) = 0$ , where the flux function $U(t, M)$ is the primitive of the prescribed velocity from the Lagrangian formulation. They utilize results from Golovaty and Nguyen [24] to handle flux functions that are not Lipschitz continuous, which is necessary for the Euler–Poisson system.

Key Contributions and Results

Equivalence Theorem: The primary result is the proof that Lagrangian solutions and entropy solutions are equivalent for the 1D pressureless Euler system. Specifically, a Lagrangian solution $X$ generates an entropy solution $M$ (its generalized inverse), and vice versa. This holds for general forcing terms $f[\rho]$ satisfying specific boundedness and continuity conditions, including the Euler–Poisson system.
Extension Beyond Blow-up: The equivalence allows the authors to define unique solutions for systems where classical solutions break down.
- Repulsive Euler–Poisson System: The paper establishes that for repulsive forces ( $\alpha < 0$ ), mass may concentrate initially but will eventually diffuse. The entropy solution concept provides a rigorous framework to describe this behavior, confirming that particles that collide due to "supercritical" initial velocities will separate, a behavior dictated by the Oleĭnik E-condition.
- Damped Systems with Confinement: For the system with linear damping and quadratic confinement (studied in [14]), the authors prove the existence of globally defined entropy solutions. They show that regardless of whether mass concentration occurs, the system asymptotically converges to a uniform mass distribution with zero velocity.
Energy Dissipation and Physicality: Through particle dynamics examples (Section 6 and 7), the paper illustrates that the entropy/Lagrangian solution is the unique "physical" continuation. In repulsive cases, alternative "sticky" solutions (where particles remain stuck after collision) violate the Oleĭnik E-condition and result in non-physical energy increases. The entropy solution ensures energy is non-increasing.
Asymptotic Stability: The paper proves that for damped systems, the solution converges exponentially fast to an equilibrium state in the $T_2$ metric. For undamped systems, a Lyapunov stability result is established, showing that solutions remain close if initial data are close.

Significance and Claims
The paper claims that the equivalence of these solution concepts is fundamental for understanding the long-time behavior of pressureless Euler systems. By linking the geometric $L^2$ -gradient flow approach with the analytical entropy solution approach, the authors provide a unified framework that:

Resolves ambiguities in solution continuation after singularity formation, particularly in repulsive regimes where particles may collide and separate.
Generalizes previous results on first-order aggregation equations (e.g., [4]) to second-order systems.
Provides a rigorous basis for studying systems with damping and confinement beyond the lifespan of classical solutions.

The authors emphasize that the one-dimensional setting is essential for this equivalence due to the unique relationship between optimal transport maps and distribution functions (generalized inverses) and the ordering of particles. They note that generalizing these results to higher dimensions is unlikely in a naive way. The work serves to unify the "sticky particle" intuition with the rigorous theory of conservation laws, offering a robust tool for analyzing complex particle dynamics in fluid models.