In this article we study an extension of Smoluchowski's discrete coagulation equation, where particle in- and output takes place. This model is frequently used to describe aggregation processes in combination with sedimentation of clusters. More precisely, we show that the evolution equation is well-posed for a large class of coagulation kernels and output rates. Additionally, in the long-time limit we prove that solutions converge to a unique equilibrium with exponential rate under a suitable smallness condition on the coefficients.
Uniqueness of mass-conserving self-similar solutions to Smoluchowski's coagulation equation is shown when the coagulation kernel $K$ is given by $K(x,x\_*)=2(x x\_*)^{-α}$, $(x,x\_*)\in (0,\infty)^2$, for some $α>0$.
It is well known that for a large class of coagulation kernels, Smoluchowski coagulation equations have particular power law solutions which yield a constant flux of mass along all scales of the system. In this paper, we prove that for some choices of the coagulation kernels there are solutions with a constant flux of mass along all scales which are not power laws. The result is proved by means of a bifurcation argument.
In the present article we introduce a variant of Smoluchowski's coagulation equation with both position and velocity variables taking a kinetic viewpoint arising as the scaling limit of a system of second-order (microscopic) coagulating particles. We focus on the rigorous study of the PDE system in the spatially-homogeneous case proving existence and uniqueness under different initial conditions in suitable weighted space, investigating also the regularity of such solutions.
The existence of self-similar solutions with fat tails for Smoluchowski's coagulation equation has so far only been established for the solvable and the diagonal kernel. In this paper we prove the existence of such self-similar solutions for continuous kernels $K$ that are homogeneous of degree $γ\in [0,1)$ and satisfy $K(x,y) \leq C (x^γ + y^γ)$. More precisely, for any $ρ\in (γ,1)$ we establish the existence of a continuous weak self-similar profile with decay $x^{-(1{+}ρ)}$ as $x \to \infty$.
The Smoluchowski coagulation-diffusion PDE is a system of partial differential equations modelling the evolution in time of mass-bearing Brownian particles which are subject to short-range pairwise coagulation. This survey presents a fairly detailed exposition of the kinetic limit derivation of the Smoluchowski PDE from a microscopic model of many coagulating Brownian particles that was undertaken in [11]. It presents heuristic explanations of the form of the main theorem before discussing the proof, and presents key estimates in that proof using a novel probabilistic technique. The survey's p…
We consider Smoluchowski's equation with a homogeneous kernel of the form $a(x,y) = x^αy ^β+ x^βy^α$ with $-1 < α\leq β< 1$ and $λ:= α+ β\in (-1,1)$. We first show that self-similar solutions of this equation are infinitely differentiable and prove sharp results on the behavior of self-similar profiles at $y = 0$ in the case $α< 0$. We also give some partial uniqueness results for self-similar profiles: in the case $α= 0$ we prove that two profiles with the same mass and moment of order $λ$ are necessarily equal, while in the case $α< 0$ we prove that two profiles with the same moments of orde…
The coagulation (or aggregation) equation was introduced by Smoluchowski in 1916 to describe the clumping together of colloidal particles through diffusion, but has been used in many different contexts as diverse as physical chemistry, chemical engineering, atmospheric physics, planetary science, and economics. The effectiveness of clumping is described by a kernel $K(x,y)$, which depends on the sizes of the colliding particles $x,y$. We consider kernels $K = (xy)^γ$, but any homogeneous function can be treated using our methods. For sufficiently effective clumping $1 \ge γ> 1/2$, the coagulat…
In this paper we consider the long time asymptotics of a linear version of the Smoluchowski equation which describes the evolution of a tagged particle moving at constant speed in a random distribution of fixed particles. The volumes $v$ of the particles are independently distributed according to a probability distribution which decays asymptotically as a power law $v^{-σ}$. The validity of the equation has been rigorously proved in \cite{NoV} for values of the exponent $σ>3$. The solutions of this equation display a rich structure of different asymptotic behaviours according to the different …
We consider two simple models for the formation of polymers where at the initial time, each monomer has a certain number of potential links (called arms in the text) that are consumed when aggregations occur. Loosely speaking, this imposes restrictions on the number of aggregations. The dynamics of concentrations are governed by modifications of Smoluchowski's coagulation equations. Applying classical techniques based on generating functions, resolution of quasi-linear PDE's, and Lagrange inversion formula, we obtain explicit solutions to these non-linear systems of ODE's. We also discuss the …
In this paper we prove the existence of a family of self-similar solutions for a class of coagulation equations with a constant flux of particles from the origin. These solutions are expected to describe the longtime asymptotics of Smoluchowski's coagulation equations with a time independent source of clusters concentrated in small sizes. The self-similar profiles are shown to be smooth, provided the coagulation kernel is also smooth. Moreover, the self-similar profiles are estimated from above and from below by $x^{-(γ+3)/2}$ as $x \to 0$, where $γ<1 $ is the homogeneity of the kernel, and ar…
Binary coagulation is an important process in aerosol dynamics by which two particles merge to form a larger one. The distribution of particle sizes over time may be described by the so-called Smoluchowski's coagulation equation. This integrodifferential equation exhibits complex non-local behaviour that strongly depends on the coagulation rate considered. We first discuss well-posedness results for the Smoluchowski's equation for a large class of coagulation kernels as well as the existence and nonexistence of stationary solutions in the presence of a source of small particles. The existence …
In this article we prove the existence of solutions to the coagulation equation with singular kernels. We use weighted L^1-spaces to deal with the singularities in order to obtain regular solutions. The Smoluchowski kernel is covered by our proof. The weak L^1 compactness methods are applied to suitably chosen approximating equations as a base of our proof. A more restrictive uniqueness result is also mentioned.
In this article we prove the existence of solutions to the singular coagulation equation with multifragmentation. We use weighted $L^1$-spaces to deal with the singularities and to obtain regular solutions. The Smoluchowski kernel is covered by our proof. The weak $L^1$ compactness methods are applied to suitably chosen approximating equations as a base of our proof. A more restrictive uniqueness result is also given.
In this paper we study the fundamental solution of the equation obtained by the linearisation of the Smoluchowski coagulation equation with the multiplicative kernel $(x y)^{λ/2}$ with $λ\in (1, 2)$ around the steady state $f(x)=x^{-(3+λ)/2}$. An explicit representation formula as well as detailed estimates on its asymptotics are obtained. We also describe in a detailed form particle fluxes between different sizes for this linearised equation.
We consider in this work a model for aggregation, where the coalescing particles initially have a certain number of potential links (called arms) which are used to perform coagulations. There are two types of arms, male and female, and two particles may coagulate only if one has an available male arm, and the other has an available female arm. After a coagulation, the used arms are no longer available. We are interested in the concentrations of the different types of particles, which are governed by a modification of Smoluchowski's coagulation equation -- that is, an infinite system of nonline…
We discuss the long-time behaviour of solutions to Smoluchowski's coagulation equation with kernels of homogeneity one, combining formal asymptotics, heuristic arguments based on linearization, and numerical simulations. The case of what we call diagonally dominant kernels is particularly interesting. Here one expects that the long-time behaviour is, after a suitable change of variables, the same as for the Burgers equation. However, for kernels that are close to the diagonal one we obtain instability of both, constant solutions and traveling waves and in general no convergence to N-waves for …
We consider mass-conserving self-similar solutions for Smoluchowski's coagulation equation with kernel $K(ξ,η)= (ξη)^λ$ with $λ\in (0,1/2)$. It is known that such self-similar solutions $g(x)$ satisfy that $x^{-1+2λ} g(x)$ is bounded above and below as $x \to 0$. In this paper we describe in detail via formal asymptotics the qualitative behavior of a suitably rescaled function $h(x)=h_λ x^{-1+2λ} g(x)$ in the limit $λ\to 0$. It turns out that $h \sim 1+ C x^{λ/2} \cos(\sqrtλ \log x)$ as $x \to 0$. As $x$ becomes larger $h$ develops peaks of height $1/λ$ that are separated by large regions wher…
We study the similarity solutions (SS) of Smoluchowski coagulation equation with multiplicative kernel $K(x,y)=(xy)^{s}$ for $s<\frac{1}{2}$. When $s<0$% , the SS consists of three regions with distinct asymptotic behaviours. The appropriate matching yields a global description of the solution consisting of a Gamma distribution tail, an intermediate region described by a lognormal distribution and a region of very fast decay of the solutions to zero near the origin. When $s\in \left( 0,\frac{1}{2}\right) $, the SS is unbounded at the origin. It also presents three regions: a Gamma distribution…
A Smoluchowski type model of coagulation in a turbulent fluid is given, first expressed by means of a stochastic model, then in a suitable scaling limit as a deterministic model with enhanced diffusion in the velocity component. A precise link between mean intensity of the turbulent velocity field and coagulation enhancement is obtained by numerical simulations, and a formula for the mean velocity difference, in agreement with the gas-kinetic model, is proved by a new method.
The asymptotic behavior of the solution of an infinite set of Smoluchowski's discrete coagulation-fragmentation-diffusion equations with non-homogeneous Neumann boundary conditions, defined in a periodically perforated domain, is analyzed. Our homogenization result, based on Nguetseng-Allaire two-scale convergence, is meant to pass from a microscopic model (where the physical processes are properly described) to a macroscopic one (which takes into account only the effective or averaged properties of the system). When the characteristic size of the perforations vanishes, the information given o…
Smoluchowski's equation is a macroscopic description of a many particle system with coagulation and shattering interactions. We give a microscopic model of the system from which we derive this equation rigorously. Provided the existence of a unique and sufficiently regular solution of Smoluchowski's equation, we prove the law of large numbers for the empirical processes. In contrast to previous derivations we assume a moderate scaling of the particle interaction, enabling us to estimate the critical fluctuation terms by using martingale inequalities. This approach can be justified in the regim…
