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Long-time asymptotics for coagulation equations with injection that do not have stationary solutions

This paper investigates the long-time asymptotic behavior of coagulation equations with a source term and homogeneous kernels where $\gamma + 2\lambda \geq 1$ , demonstrating that mass transport is driven by collisions between particles of different sizes and establishing the existence of self-similar solutions for this regime while proving their non-existence in the complementary case.

Original authors: Iulia Cristian, Marina A. Ferreira, Eugenia Franco, Juan J. L. Velázquez

Published 2026-07-22

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CC BY 4.0

Original authors: Iulia Cristian, Marina A. Ferreira, Eugenia Franco, Juan J. L. Velázquez

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 a world made entirely of tiny, floating Lego bricks. Some are single bricks, some are tiny towers, and others are massive, sprawling castles. In the atmosphere, clouds, and even in industrial smoke, these "bricks" (called clusters or particles) are constantly bumping into each other. When they collide, they stick together to form something bigger. This process is called coagulation. It's the reason fog forms, why soot clumps up in your car's exhaust, and how clouds grow heavy enough to rain. Scientists use math to predict how these clumps grow over time. Usually, if you keep adding new tiny bricks into the mix (a "source"), the system eventually settles into a steady rhythm where the number of small, medium, and large clumps stays constant. It's like a busy factory where the rate of new bricks arriving perfectly matches the rate of them being glued together into giant structures.

But what happens if the rules of the game change? What if the way the bricks stick together is so aggressive that the tiny ones get swallowed up by the giants almost instantly, faster than the factory can keep up? This is the puzzle tackled in a new paper by mathematicians Iulia Cristian, Marina A. Ferreira, Eugenia Franco, and Juan J. L. Velázquez. They are looking at a specific, tricky scenario where the "glue" between particles of very different sizes is incredibly strong. In this world, the usual steady rhythm breaks down. Instead of a calm, steady state, the system goes into a wild, self-similar dance where everything scales up together as time goes on. The authors wanted to know: Does this wild dance have a pattern? Can we describe it with a neat mathematical formula, or does the system behave in a way that defies description?

The Great Glue-Up: When Tiny Particles Vanish

The story begins with a simple question: If you keep pouring tiny particles (like dust or smoke) into a room where they stick together, what does the room look like after a very long time?

In many cases, the answer is a "stationary solution." Think of this as a balanced ecosystem. You have a steady stream of new tiny particles coming in, and a steady stream of them merging into larger ones. The number of small, medium, and large particles stays roughly the same every day. It's a stable, predictable world.

However, the authors discovered that this stability only exists if the "glue" between particles isn't too strong. Specifically, they looked at a mathematical rule involving two numbers, $\gamma$ and $\lambda$ , which describe how the stickiness changes with size. If the combination of these numbers is small ( $\gamma + 2\lambda < 1$ ), the system finds its balance. The tiny particles hang around long enough to be part of the steady mix.

But if the glue is strong enough ( $\gamma + 2\lambda \ge 1$ ), the balance breaks. In this regime, the tiny particles don't just merge; they get snatched up by the giants almost instantly. It's as if the tiny Lego bricks are so eager to join the big castles that they disappear from the floor the moment they are dropped. Because they vanish so fast, you can never reach a steady state where the number of tiny bricks stays constant. The system is in a constant state of flux, rushing toward infinity.

The Search for a Pattern in the Chaos

Since the system doesn't settle down, the authors asked: Does it at least follow a predictable pattern as it rushes forward? In math, this is called a "self-similar solution." Imagine taking a photo of the particle distribution at one time, and then another photo a year later. If the system is self-similar, the second photo is just a zoomed-out, stretched version of the first one. The shape of the distribution stays the same; only the scale changes.

The authors set out to prove whether such a pattern exists for this "super-glue" scenario. They built a new mathematical model that includes a special "transport term." You can think of this term as a conveyor belt. Because the tiny particles are being sucked into the giants so quickly, the model treats them as if they are being instantly transported from the "tiny" zone to the "huge" zone. This conveyor belt is driven by the injection of new particles.

The Big Discovery: The "Minimal Size" and the Dead Zone

The paper's main finding is a mix of "Yes, but..." and "No, never."

The "Yes" (for some cases):
The authors proved that if the exponent $\gamma$ is greater than $-1$, a self-similar pattern does exist. But it has a very strange and fascinating feature: a "dead zone."

Imagine the conveyor belt doesn't just move particles; it skips a whole section of the floor. The authors found that for large times, there is a "minimal cluster size." Below this size, the number of particles is exactly zero.

The Analogy: Think of a river flowing into a giant whirlpool. The water doesn't just swirl slowly; it gets sucked in so fast that there is a dry patch of riverbed right before the whirlpool. No water exists in that dry patch. Similarly, in this math model, there is a specific size threshold. Any particle smaller than this threshold is gone, instantly transported to the larger sizes. The distribution of particles starts at this "minimal size" and then tapers off.
The Shape: For particles larger than this threshold, the distribution follows a specific curve that decays very quickly (exponentially). It's like a hill that rises sharply from the dry patch and then drops off into a cliff.

The "No" (for other cases):
The authors also proved that if the exponent $\gamma$ is less than or equal to $-1$, this self-similar pattern does not exist.

The Reason: In this scenario, the math shows that the system would have to behave in a way that violates the basic laws of mass conservation. It's like trying to build a tower where the bricks keep multiplying out of thin air or disappearing into a black hole in a way that breaks the rules of physics. The authors showed that no matter how you try to arrange the particles, you cannot find a stable, repeating pattern. The system simply doesn't have a "shape" it can settle into.

Why This Matters

This isn't just about abstract numbers. The authors point out that these "super-glue" conditions appear in real-world atmospheric science. For example, when studying how aerosols (tiny pollution particles) grow in the atmosphere, understanding whether they form a steady cloud or a rapidly shifting, self-similar wave is crucial.

The paper confirms that in certain extreme conditions, the atmosphere doesn't just "settle." Instead, it undergoes a rapid, organized transformation where tiny particles vanish from existence to feed the growth of massive ones, leaving a distinct "gap" in the sizes of particles present. The authors have rigorously proved that this gap exists and that the pattern of growth is predictable in some cases, while in others, the system is too chaotic to have a simple pattern at all.

In short, the paper tells us that when the glue is strong enough, the universe of particles doesn't just grow; it reorganizes itself into a new, dynamic structure with a clear "forbidden zone" for small sizes, and this structure is mathematically real and proven, provided the rules of the game aren't too extreme.

Problem Statement
This paper investigates the long-time asymptotic behavior of coagulation equations with a source term that injects clusters of size order one. The study focuses on the regime where the coagulation kernel $K(x, y)$ is homogeneous of degree $\gamma < 1$ and satisfies specific polynomial bounds involving a parameter $\lambda$ . The analysis is restricted to the case where $\gamma + 2\lambda \geq 1$ , a range where the transport of mass toward infinity is driven by collisions between particles of significantly different sizes.

In contrast to the case $\gamma + 2\lambda < 1$ , where stationary non-equilibrium solutions exist for the equation with a source, the authors note that for $\gamma + 2\lambda \geq 1$ , no stationary solution to the equation $K[f] + \eta = 0$ exists. Consequently, the standard approach of converging to a stationary profile fails. The central question is whether the solutions to the time-dependent equation with injection converge to a self-similar profile as $t \to \infty$ , and if so, under what conditions on $\gamma$ and $\lambda$ .

Methodology
The authors employ a combination of formal asymptotic analysis and rigorous functional analysis:

Formal Asymptotics: The paper first derives an approximate equation for the distribution of large clusters ( $x \gg 1$ ). By analyzing the interaction between small injected particles and large clusters, the coagulation operator is approximated by a non-local transport term. This leads to a reduced equation (1.16) governing the evolution of the outer distribution, which includes a transport term driven by the injection rate and the moments of the distribution.
Self-Similar Ansatz: The reduced equation is shown to admit self-similar solutions of the form $f_s(t, x) = t^{-\frac{3+\gamma}{1-\gamma}} \Phi(x t^{-\frac{2}{1-\gamma}})$ . The problem is reduced to finding a stationary profile $\Phi$ satisfying a specific integro-differential equation (1.19).
Existence Proof (for $\gamma > -1$ ):
- The authors utilize a truncation and regularization technique, introducing a truncated kernel $K_R$ to handle the singularities and unbounded domains.
- They construct a time-dependent solution for the truncated equation using Lagrangian coordinates and a fixed-point argument (Tychonoff fixed point theorem) within an invariant set of measures.
- They establish uniform bounds on moments ( $M_1$ , $M_{\gamma+\lambda}$ , $M_{2-\gamma-\lambda}$ ) and prove the invariance of a specific set of measures under the evolution semigroup.
- By passing to the limit as the truncation parameter $R \to \infty$ , they extract a weak limit $\Phi$ that satisfies the self-similar profile equation.
Non-Existence Proof (for $\gamma \leq -1$ ):
- The proof proceeds by contradiction. Assuming a self-similar profile exists, the authors analyze the properties of the flux and the support of the solution.
- They demonstrate that for $\gamma + 2\lambda > 1$ , any solution must vanish in a neighborhood of the origin.
- By testing the profile equation with the constant function $\phi \equiv 1$ , they derive a contradiction regarding the sign of the total mass evolution, thereby proving non-existence.
- For the critical case $\gamma + 2\lambda = 1$ , they prove non-existence under the additional assumption that the moment $\int_0^1 x^{-\lambda} \Phi(dx)$ is finite.

Key Contributions and Results

Existence of Self-Similar Profiles: The paper proves that if $\gamma > -1$ and $\gamma + 2\lambda \geq 1$ (within the non-gelling regime $\gamma < 1, \gamma + \lambda < 1$ ), there exists at least one self-similar profile $\Phi$ for the coagulation equation with transport.
Non-Existence Results:
- If $\gamma \leq -1$ and $\gamma + 2\lambda > 1$ , no self-similar profile exists.
- If $\gamma \leq -1$ and $\gamma + 2\lambda = 1$ , no self-similar profile exists that satisfies the condition $\int_0^1 x^{-\lambda} \Phi(dx) < \infty$ .
Structure of the Solution (Gap Phenomenon): A significant finding is that the constructed self-similar profiles $\Phi$ vanish identically on an interval $(0, \rho(M_{\gamma+\lambda}))$ , where $\rho$ depends on the moment $M_{\gamma+\lambda}$ . This implies that for large times, the injected particles are transferred almost instantaneously to clusters with sizes $x \geq \rho(M_{\gamma+\lambda}) t^{\frac{2}{1-\gamma}}$ . The region of small cluster sizes becomes negligible.
Decay Properties: The authors prove that the density of the self-similar profile decays exponentially as the cluster size goes to infinity, similar to profiles found in other coagulation contexts.
Regularity: The constructed profiles are shown to be absolutely continuous with respect to the Lebesgue measure.

Significance and Claims
The paper claims to provide a complete description of the long-time asymptotics for coagulation equations with injection in the regime $\gamma + 2\lambda \geq 1$ , a case where stationary solutions do not exist. The primary significance lies in:

Identifying the Transport Mechanism: The work rigorously justifies the approximation of the coagulation operator by a transport term when collisions between disparate sizes dominate, leading to a new class of self-similar solutions.
The "Minimal" Cluster Size: The discovery of a "minimal" cluster size below which the density is zero for large times is highlighted as a remarkable feature not previously observed in the literature on self-similar solutions for coagulation equations. This contrasts sharply with the behavior in the $\gamma + 2\lambda < 1$ regime.
Sharp Thresholds: The paper establishes precise thresholds for the existence of these solutions based on the parameters $\gamma$ and $\lambda$ , specifically distinguishing the case $\gamma > -1$ from $\gamma \leq -1$ .

The authors note that their results for the critical case $\gamma + 2\lambda = 1$ are more fragmentary than for the super-critical case, particularly regarding the non-existence of solutions without the additional moment condition. They also state that the long-time asymptotics for the case $\gamma \leq -1$ where self-similar solutions do not exist remain an open question for future work.

The Great Glue-Up: When Tiny Particles Vanish

The Search for a Pattern in the Chaos

The Big Discovery: The "Minimal Size" and the Dead Zone

Why This Matters

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