Coagulation-fragmentation Equations Research Articles

Coagulation–Fragmentation Equilibrium for Charged Dust: Abundance of Submicron Grains Increases Dramatically in Protoplanetary Disks

Dust coagulation in protoplanetary disks is not straightforward and is subject to several slowdown mechanisms, such as bouncing, fragmentation, and radial drift to the star. Furthermore, dust grains in UV-shielded disk regions are negatively charged due to collisions with the surrounding electrons and ions, which leads to their electrostatic repulsion. For typical disk conditions, the relative velocities between micron-sized grains are small, and their collisions are strongly affected by the repulsion. On the other hand, collisions between pebble-sized grains can be too energetic, leading to grain fragmentation. The aim of the present paper is to study the combined effect of the electrostatic and fragmentation barriers on dust evolution. We numerically solve the Smoluchowski coagulation–fragmentation equation for grains whose charging occurs under conditions typical for the inner disk regions, where thermal ionization operates. We find that dust fragmentation efficiently resupplies the population of small grains under the electrostatic barrier. As a result, the equilibrium abundance of submicron grains is enhanced by several orders of magnitude compared to the case of neutral dust. For some conditions with fragmentation velocities of ∼1 m s−1, macroscopic grains are completely destroyed.

Grand canonical Brownian dynamics simulations of adsorption and self-assembly of SAS-6 rings on a surface.

The Spindle Assembly Abnormal Protein 6 (SAS-6) forms dimers, which then self-assemble into rings that are critical for the nine-fold symmetry of the centriole organelle. It has recently been shown experimentally that the self-assembly of SAS-6 rings is strongly facilitated on a surface, shifting the reaction equilibrium by four orders of magnitude compared to the bulk. Moreover, a fraction of non-canonical symmetries (i.e., different from nine) was observed. In order to understand which aspects of the system are relevant to ensure efficient self-assembly and selection of the nine-fold symmetry, we have performed Brownian dynamics computer simulation with patchy particles and then compared our results with the experimental ones. Adsorption onto the surface was simulated by a grand canonical Monte Carlo procedure and random sequential adsorption kinetics. Furthermore, self-assembly was described by Langevin equations with hydrodynamic mobility matrices. We find that as long as the interaction energies are weak, the assembly kinetics can be described well by coagulation-fragmentation equations in the reaction-limited approximation. By contrast, larger interaction energies lead to kinetic trapping and diffusion-limited assembly. We find that the selection of nine-fold symmetry requires a small value for the angular interaction range. These predictions are confirmed by the experimentally observed reaction constant and angle fluctuations. Overall, our simulations suggest that the SAS-6 system works at the crossover between a relatively weak binding energy that avoids kinetic trapping and a small angular range that favors the nine-fold symmetry.

Local mass-conserving solution for a critical coagulation-fragmentation equation

The critical coagulation-fragmentation equation with multiplicative coagulation and constant fragmentation kernels is known to not have global mass-conserving solutions when the initial mass is greater than 1. We show that for any given positive initial mass with finite second moment, there is a time T⁎>0 such that the equation possesses a unique mass-conserving solution up to T⁎. The novel idea is to singularly perturb the constant fragmentation kernel by small additive terms and study the limiting behavior of the solutions of the perturbed system via the Bernstein transform.

Well-posedness of the coagulation-fragmentation equation with size diffusion

Local and global well-posedness of the coagulation-fragmentation equation with size diffusion are investigated. Owing to the semilinear structure of the equation, a semigroup approach is used, building upon generation results previously derived for the linear fragmentation-diffusion operator in suitable weighted $L_1$-spaces.

An application of semigroup theory to the coagulation-fragmentation models

We present the existence and uniqueness of strong solutions for the continuous coagulation-fragmentation equation with singular fragmentation and essentially bounded coagulation kernel using semigroup theory of operators. Initially, we reformulate the coupled coagulation-fragmentation problem into the semilinear abstract Cauchy problem (ACP) and consider it as the nonlinear perturbation of the linear fragmentation operator. The existence of the substochastic semigroup is proved for the pure fragmentation equation. Using the substochastic semigroup and some related results for the pure fragmentation equation, we prove the existence of global nonnegative, strong solution for the coagulation-fragmentation equation.

A structured coagulation-fragmentation equation in the space of radon measures: Unifying discrete and continuous models

We present a structured coagulation-fragmentation model which describes the population dynamics of oceanic phytoplankton. This model is formulated on the space of Radon measures equipped with the bounded Lipschitz norm and unifies the study of the discrete and continuous coagulation-fragmentation models. We prove that the model is well-posed and show it can reduce down to the classic discrete and continuous coagulation-fragmentation models. To understand the interplay between the physical processes of coagulation and fragmentation and the biological processes of growth, reproduction, and death, we establish a regularity result for the solutions and use it to show that stationary solutions are absolutely continuous under some conditions on model parameters. We develop a semi-discrete approximation scheme which conserves mass and prove its convergence to the unique weak solution. We then use the scheme to perform numerical simulations for the model.

Accurate and efficient approximations for generalized population balances incorporating coagulation and fragmentation

This study focuses on development of two approaches based on finite volume schemes for solving both one-dimensional and multidimensional nonlinear simultaneous coagulation-fragmentation population balance equations (PBEs). Existing finite volume schemes and sectional methods such as fixed pivot technique and cell average technique have many issues related to accuracy and efficiency. To resolve these challenges, two finite volume schemes are developed and compared with the cell average technique along with the exact solutions. The new schemes have features such as simpler mathematical formulations, easy to code and robust to apply on nonuniform grids. The numerical testing shows that both new finite volume schemes compute the number density functions and their corresponding integral moments with higher precision on a coarse grid by consuming lesser CPU time. In addition, both schemes are extended to approximate generalized simultaneous coagulation-fragmentation problems and retains the numerical accuracy and efficiency. For the higher dimensional PBEs (2D and 3D), the investigation and verification of the numerical schemes is done by deriving new exact integral moments for various combinations of coagulation kernels, selection functions and fragmentation kernels.

Solutions with peaks for a coagulation-fragmentation equation. Part II: Aggregation in peaks

The aim of this two-part paper is to investigate the stability properties of a special class of solutions to a coagulation-fragmentation equation. We assume that the coagulation kernel is close to the diagonal kernel, and that the fragmentation kernel is diagonal. In a companion paper we constructed a two-parameter family of stationary solutions concentrated in Dirac masses, and we carefully studied the asymptotic decay of the tails of these solutions, showing that this behaviour is stable. In this paper we prove that for initial data which are sufficiently concentrated, the corresponding solutions approach one of these stationary solutions for large times.

Existence and uniqueness of steady-state solution to a singular coagulation–fragmentation equation

In this article, we prove the existence of an equilibrium solution to the coagulation–fragmentation equation with •[(i)] a singular coagulation kernel in the form K(x,y)=(1+xλ+yλ)(xy)σ,where σ∈0,12 and λ−σ∈[0,1], and•[(ii)] a fragmentation kernel in the form F(x,y)=b0+b(x+y)m,where b0>0 and b,m≥0. Importantly, in the entire study, the detailed balance condition is not assumed. It is shown that the equilibrium solution of the problem under consideration is unique in the space of all continuous functions on (0,∞). The proposed results are demonstrated by a numerical illustration.

Uniqueness of solutions to the coagulation–fragmentation equation with singular kernel

The existence of a solution to an important singular coagulation equation with a multiple fragmentation kernel has been recently proved in Jpn J Ind Appl Math 35(3):1283–1302, 2018. This paper proves the uniqueness of the solution to the same problem in the function space $$\varOmega _{.,r_2} (T) = \bigcup _{\lambda >0 }\varOmega _{\lambda , r_2} (T)$$, where $$\varOmega _{\lambda , r_2} (T)$$ is the space of all continuous functions f such that $$\begin{aligned} \Vert f\Vert _{\lambda , r_2} : = \sup \limits _{0\le t \le T} \int _0^{\infty } \left( \exp (\lambda x ) + \frac{1}{x^{r_2}}\right) |f(x,t)| dx ~<~ \infty \end{aligned}$$and $$0< r_2 < 1$$.

Temporal oscillations in Becker–Döring equations with atomization

We prove that time-periodic solutions arise via Hopf bifurcation in a finite closed system of coagulation-fragmentation equations. The system we treat is a variant of the Becker–Döring equations, in which clusters grow or shrink by addition or deletion of monomers. To this is added a linear atomization reaction for clusters of maximum size. The structure of the system is motivated by models of gas evolution oscillators in physical chemistry, which exhibit temporal oscillations under certain input/output conditions.

Well-posedness for boundary value problems for coagulation-fragmentation equations

We investigate a coagulation-fragmentation equation with boundary data, establishing the well-posedness of the initial value problem when the coagulation kernels are bounded at zero and showing existence of solutions for the singular kernels relevant in the applications. We determine the large time asymptotic behavior of solutions, proving that solutions converge exponentially fast to zero in the absence of fragmentation and stabilize toward an equilibrium if the boundary value satisfies a detailed balance condition. Incidentally, we obtain an improvement in the regularity of solutions by showing the finiteness of negative moments for positive time.

Mass-conserving solutions to coagulation-fragmentation equations with balanced growth

A specific class of coagulation and fragmentation coefficients is considered for which the strength of coagulation is balanced by that of multiple fragmentation. Existence and uniqueness of mass-conserving solutions is proved when the initial total mass i

Global solutions of continuous coagulation–fragmentation equations with unbounded coefficients

In this paper we prove the existence of global classical solutions to continuous coagulation–fragmentation equations with unbounded coefficients under the sole assumption that the coagulation rate is dominated by a power of the fragmentation rate, thus improving upon a number of recent results by not requiring any polynomial growth bound for either rate. This is achieved by proving a new result on the analyticity of the fragmentation semigroup and then using its regularizing properties to prove the local and then, under a stronger assumption, the global classical solvability of the coagulation–fragmentation equation considered as a semilinear perturbation of the linear fragmentation equation. Furthermore, we show that weak solutions of the coagulation–fragmentation equation, obtained by the weak compactness method, coincide with the classical local in time solutions provided the latter exist.

Solutions with peaks for a coagulation–fragmentation equation. Part I: stability of the tails

The aim of this two-part paper is to investigate the stability properties of a special class of solutions to a coagulation–fragmentation equation. We assume that the coagulation kernel is close to the diagonal kernel, and that the fragmentation kernel is diagonal. We construct a two-parameter family of stationary solutions concentrated in Dirac masses. We carefully study the asymptotic decay of the tails of these solutions, showing that this behavior is stable. In a companion paper, we prove that for initial data which are sufficiently concentrated, the corresponding solutions approach one of these stationary solutions for large times.

Stationary solutions to coagulation-fragmentation equations

Existence of stationary solutions to the coagulation-fragmentation equation is shown when the coagulation kernel K and the overall fragmentation rate a are given by K(x,y)=xαyβ+xβyα and a(x)=xγ, respectively, with 0≤α≤β≤1, α+β∈[0,1), and γ>0. The proof requires two steps: a dynamical approach is first used to construct stationary solutions under the additional assumption that the coagulation kernel and the overall fragmentation rate are bounded from below by a positive constant. The general case is then handled by a compactness argument.

Mass-conserving self-similar solutions to coagulation–fragmentation equations

Existence of mass-conserving self-similar solutions with a sufficiently small total mass is proved for a specific class of homogeneous coagulation and fragmentation coefficients. The proof combines a dynamical approach to construct such solutions for a regularized coagulation–fragmentation equation in scaling variables and a compactness method.

A High Order Positivity Preserving DG Method for Coagulation-Fragmentation Equations

We design, analyze, and numerically validate a novel discontinuous Galerkin (DG) method for solving the coagulation-fragmentation equations. The DG discretization is applied to the conservative form of the model, with flux terms evaluated by Gaussian quadrature with $Q=k+1$ quadrature points for polynomials of degree $k$. The first moment (total mass) is naturally conserved by the scheme construction, and the positivity of the mass density is enforced by the use of a scaling limiter based on positive cell averages. The positivity of cell averages is shown to propagate by the time discretization, provided a proper time step restriction is imposed.

The discrete unbounded coagulation-fragmentation equation with growth, decay and sedimentation

In this paper we study the discrete coagulation–fragmentation models with growth, decay and sedimentation. We demonstrate the existence and uniqueness of classical global solutions provided the linear processes are sufficiently strong. This paper extends several previous results both by considering a more general model and and also signnificantly weakening the assumptions. Theoretical conclusions are supported by numerical simulations.

Existence of mass conserving solution for the coagulation–fragmentation equation with singular kernel

In this article, we investigate the existence of a solution to the initial value problem associated to a singular coagulation–fragmentation equation. We analyze the problem with singular coagulation kernel and multiple fragmentation kernel. It is also shown that the solution satisfies the mass conservation law.