Coagulation Operators: Theory and Applications

• Coagulation operators are nonlinear mappings defined by bilinear forms that merge particles and clusters in kinetic models.
• They underpin key dynamical phenomena such as mass conservation, gelation, and critical clustering in both discrete and continuous systems.
• Analytical and probabilistic techniques combined with functional analysis provide existence, uniqueness, and explicit solution strategies in various models.

Coagulation operators are fundamental nonlinear constructs governing the binary or multiple merging of particles, clusters, or combinatorial structures in a variety of mathematical and physical systems. Their rigorous analysis is central in the study of kinetic equations such as the Smoluchowski coagulation equation, spatially inhomogeneous and mass-structured models, growth-fragmentation-coagulation systems, discrete particle systems, random partitions, and specialized variants like limited-aggregation polymerization. Coagulation operators underpin both the analytic properties and probabilistic interpretations of aggregation dynamics, including mass conservation, gelation, clustering, and critical behavior.

1. Analytical Definition and Structures

The paradigmatic form of the coagulation operator arises in the Smoluchowski equation for number densities $f(t,x)$ of particles of mass $x$: $$\partial_t f(t,x) = \frac12 \int_0^x K(x-y, y) f(t, x-y) f(t, y) dy - f(t, x) \int_0^\infty K(x, y) f(t, y) dy,$$ where $K(x, y)$ is a symmetric coagulation kernel (Ferreira, 2020). More generally, the operator is characterized as a bilinear mapping on measures $f$ via

$$\langle\varphi, L[f]\rangle = \frac12 \iint K(x, y)[\varphi(x + y) - \varphi(x) - \varphi(y)] f(dx) f(dy)$$

for test functions $\varphi$.

In growth-fragmentation-coagulation systems, the operator appears as

$$(\mathcal{K} f)(x) = \frac12 \int_0^x k(x - y, y) f(x - y) f(y) dy - f(x)\int_0^\infty k(x, y) f(y) dy,$$

and is incorporated into extended semilinear PDEs with additional drift and fragmentation components (Banasiak et al., 2020).

In discrete settings (particle sizes as integers), the coagulation operator $Q_c$ on a weighted sequence space $\ell^1_w$ is given for $x = (x_i)_{i \geq 1}$ as

$$(Q_c(x))_k = \frac12 \sum_{i+j=k} K(i, j) x_i x_j - x_k \sum_{j \geq 1} K(k, j) x_j$$

(Kerr et al., 30 Apr 2025). Analogs operate for mass partitions and combinatorial random structures (Ho et al., 2022).

2. Coagulation Kernels and Admissibility

The kernel $K(x, y)$ determines not only the physics but also the mathematical properties of the evolution. Kernels commonly satisfy

• Symmetry: $K(x, y) = K(y, x) \geq 0$,
• Power-law bounds: $$c_1[x^{\lambda+\gamma} y^{-\lambda} + y^{\lambda+\gamma} x^{-\lambda}] \leq K(x, y) \leq c_2[x^{\lambda+\gamma} y^{-\lambda} + y^{\lambda+\gamma} x^{-\lambda}]$$ for suitable $c_1, c_2, \lambda, \gamma$ (Ferreira, 2020).

Specific forms include:

• The additive kernel $K(x, y) = x + y$ (solvable, related to the Borel law) (İşeri et al., 2016),
• The constant kernel $K \equiv 2$ (explicit generating function solutions) (Ferreira, 2020),
• Physically derived kernels, e.g., Smoluchowski, Brownian, or Ornstein-Uhlenbeck collision rates (Li, 2014):

Model	Kernel Formula	Features
Classical Brownian	$(x^{1/3}+y^{1/3})(x^{-1/3}+y^{-1/3})$	Standard for colloidal spheres
OU (inertial)	$\sqrt{b(x)+b(y)}(r(x)+r(y))^{d-1}$	Velocity-dependent, non-diffusive
Limited aggregation	$K((a, m), (a', m')) = a \cdot a'$	Arms-constrained, env. for polymers

Non-monotone, unbounded, or highly singular kernels may require specialized moment bounds and regularization, especially in high-mass regimes (Banasiak et al., 2020).

3. Functional-Analytic and Mapping Properties

Coagulation operators are locally Lipschitz and quadratic on appropriate weighted Banach spaces. For continuous systems, they map $X_{0,m}$ to $X_{0,p}$ (weighted $L^1$ spaces), are Fréchet-differentiable, and their boundedness is controlled by moments of $f$ and the kernel growth exponents: $$\|\mathcal{K}(f, f)\|_{0, p} \leq C\bigl(\|f\|_{0, m} + \|f\|_{0, m}^2\bigr)$$ for $m > \gamma + \alpha$ when $k(x, y) \leq k_0(1 + x^\gamma)(1 + y^\alpha)$ (Banasiak et al., 2020).

For discrete coagulation equations, $Q_c$ is bilinear, locally Lipschitz on $\ell^1_w$, and quadratic: $\|Q_c(x)\|_w \leq C\|x\|_w^2,$ with boundedness and Lipschitz constants depending on time-compactness and local growth rates of $K(i, j)$ (Kerr et al., 30 Apr 2025).

Spatially inhomogeneous variants embed the operator in Banach spaces of equicontinuous kernels, interacting with contraction semigroups generated by drift-diffusion operators (Bailleul, 2010).

4. Key Models, Explicit Solutions, and Physical Contexts

A foundational setting is the Smoluchowski equation, which models aerosol dynamics, colloidal coagulation, and particle clustering. In the presence of constant or additive kernels, explicit solutions are available via generating function (e.g., Borel distribution for $K(x, y) = x + y$ (İşeri et al., 2016); exponential profile for $K \equiv 2$ (Ferreira, 2020)). Multi-component problems admit vectorial extensions, with localization of mass along composition diagonals.

Coagulation in stochastic combinatorial systems (exchangeable partitions, Poisson-Dirichlet distributions) is described by composite operators acting on bridges of mass partitions (Ho et al., 2022). “Dual dependent coagulation” extends classic Poisson-Dirichlet operations to the full family of stable Poisson-Kingman models, with explicit structure for bridge composition, mixing densities, and joint laws.

Physical derivations link the kernel structure to microscopic dynamics: Brownian motion yields kernels scaling with diffusivity, whereas inertial effects (Ornstein-Uhlenbeck dynamics) reduce aggregation rates for small particles (Li, 2014).

Specialized models such as coagulation with limited aggregations encode physical or chemical constraints—e.g., finite valence in polymerization—via structurally modified operators and systematics relating to random configuration models and hydrodynamic limits (Bertoin, 2012).

5. Well-posedness, Blow-up, and Critical Phenomena

Rigorous existence and uniqueness for coagulation equations depend on kernel bounds and initial moment controls. Key results:

• If $\gamma < 1$ in $K(x, y) \leq c_2(x^\gamma + y^\gamma)$ with suitable initial $x^2$-moment, mass-conserving solutions exist globally in time (Ferreira, 2020).
• Moment regularization in the linear growth-fragmentation semigroup allows solvability for unbounded kernels, provided growth exponents satisfy $\gamma, \alpha < y_0$ (Banasiak et al., 2020).
• For bounded spatial kernels and strongly dissipative drift, global well-posedness is ensured for small initial data (Bailleul, 2010).
• Criteria for stationary solutions under source terms depend on the “sub-critical window” $|\gamma + 2\lambda| < 1$; outside this, no steady solution exists (Ferreira, 2020).

Gelation—finite-time loss of finite mass to an infinite particle—occurs for kernels with sufficiently high homogeneity. Models with aggregation limits can avoid gelation entirely, or admit exactly computable post-gelation states (Bertoin, 2012).

6. Extensions: Multi-component, Random Graphs, and Dual Operators

Complex chemical or compositional structure is incorporated via vector-valued coagulation operators and multi-indexed sizes. Operator-theoretic results (mapping, differentiability, and regularity) generalize naturally to these systems (Ferreira, 2020).

In random configuration models, the operator admits a probabilistic interpretation: the distribution of rooted clusters in large random graphs converges to the total progeny of a Galton–Watson process, fully governed by arm-limited or size-dependent coagulation (Bertoin, 2012). Duality with fragmentation and deep connections to Lee-Yang/Pitman–Yor processes are explicit in the probabilistic structure of some coagulation operators (Ho et al., 2022).

7. Summary Table of Coagulation Operator Forms

Model/Class	Operator (gain–loss)	Physical/Mathematical Context
Classical Smoluchowski	$$\frac12\int K(x-y, y) f(x-y) f(y) dy - f(x)\int K(x, y) f(y)dy$$	Aerosols, colloids, polymer gels (Ferreira, 2020)
Weighted Discrete	$$\frac12\sum_{i+j=k} K(i, j) x_i x_j - x_k\sum_j K(k, j) x_j$$	Clusters of discrete size/mass (Kerr et al., 30 Apr 2025)
Growth–Fragmentation	As above, embedded with fragmentation, drift, and absorption	Cell division, polymer breakup (Banasiak et al., 2020)
Random Mass Partition	Composition of bridges, $\mathrm{COAG}_Q(\widetilde{V})$	Exchangeable partitions, PD laws (Ho et al., 2022)
Limited Aggregation	$K((a, m), (a', m')) = a a'$	Polymers with finite valence (Bertoin, 2012)
Spatial Coagulation	Quadratic on measures in $x$ and $y$, with spatial drift/diff.	Dispersed particle systems (Bailleul, 2010)

• (İşeri et al., 2016) Depinning as a coagulation process
• (Banasiak et al., 2020) Growth-fragmentation-coagulation equations with unbounded coagulation kernels
• (Kerr et al., 30 Apr 2025) Discrete coagulation–fragmentation systems in weighted $\ell^1$ spaces
• (Norris, 2014) Measure solutions for the Smoluchowski coagulation-diffusion equation
• (Ho et al., 2022) Inverse clustering of Gibbs Partitions via independent fragmentation and dual dependent coagulation operators
• (Li, 2014) Effects of microscopic dynamics on Brownian coagulation
• (Ferreira, 2020) Coagulation equations for aerosol dynamics
• (Bailleul, 2010) Spatial coagulation with bounded coagulation rate
• (Bertoin, 2012) Coagulation with limited aggregations