CCoagulation equations for aerosol dynamics
Marina A. FerreiraSeptember 10, 2020
Abstract
Binary coagulation is an important process in aerosol dynamics by which two par-ticles merge to form a larger one. The distribution of particle sizes over time may bedescribed by the so-called Smoluchowski’s coagulation equation. This integrodifferentialequation exhibits complex non-local behaviour that strongly depends on the coagulationrate considered. We first discuss well-posedness results for the Smoluchowski’s equationfor a large class of coagulation kernels as well as the existence and nonexistence of sta-tionary solutions in the presence of a source of small particles. The existence result usesSchauder fixed point theorem, and the nonexistence result relies on a flux formulationof the problem and on power law estimates for the decay of stationary solutions with aconstant flux. We then consider a more general setting. We consider that particles maybe constituted by different chemicals, which leads to multi-component equations describ-ing the distribution of particle compositions. We obtain explicit solutions in the simplestcase where the coagulation kernel is constant by using a generating function. Using anapproximation of the solution we observe that the mass localizes along a straight line inthe size space for large times and large sizes.
Contents a r X i v : . [ m a t h - ph ] S e p Perspectives and open problems 20
We consider particle systems where moving particles undergo binary coagulation, forminglarger particles. This simple system can be used to study the dynamics of aerosols in theatmosphere [17] as well as raindrop formation, smoke, sprays and galaxies [1, 9].When the number of particles is very large it becomes more relevant to study the averagebehaviour of the particles rather than individual particle behaviour. This motivates a statis-tical description of the system. In 1916 Smoluschowski [33] proposed an equation to describethe particle size distribution over time, assuming that the system is homogeneous in space.Let f ( x, t ) be the density of particles of size x > t >
0. The Smoluchowski’scoagulation equation, or simply coagulation equation, is the following mean-field equation forthe evolution of f∂ t f ( x, t ) = 12 (cid:90) x K ( x − y, y ) f ( x − y ) f ( x ) dy − (cid:90) ∞ K ( x, y ) f ( x ) f ( y ) (1.1)where K ( x, y ) is the coagulation rate between particles of size x and y . The first term onthe right hand-side is the gain term due to the coagulation between particles of size x − y and particles of size x to create a particle of size x . The second term is the loss term due tothe coagulation of particles of size x with any other particle in the system. Equation (1.1) isan integrodifferential equation belonging to the class of kinetic equations as well as Generaldynamic equations.If the size is an integer, for example, if the size is described by the number of moleculesthat constitute a particle, then the density of particles of size α ∈ N + at time t , n α ( t ), satisfiesthe discrete Smoluchowski’s coagulation, ∂ t n α = 12 (cid:88) β<α K α − β,β n α − β n β − n α (cid:88) β> K α,β n β . (1.2)Complementary research lines have emerged over the last decades on experimental, nu-merical and theoretical issues. Algorithms to simulate Smoluschowski’s coagulation equationhave been developed to test hypotheses drawn from atmospheric data [25, 35] (see [26] fora survey on numerical methods). On the other hand, theoretical results have clarified is-sues such as existence and uniqueness of solutions for general classes of kernels [31, 16] orthe behaviour of the solutions for explicitly solvable kernels [30] as well as general kernels[4, 5, 12, 15].A particle may be characterized not only by its size but also by its composition, leadingto multicomponent coagulation equations where the size is described by a vector x ∈ R d + (or α ∈ N d ) representing the size of each of the chemical components of a particle. An applicationof multicomponent equations to aerosol dynamics is described in Section 2.2.In this paper we review analytic results related to the continuous onecomponent equation(1.1) and to the discrete multicomponent equation (1.2), with α, β ∈ N d . We start with apreliminary Section 2 where we give a short overview on various topics related to propertiesof the solutions, applications, derivation from particle systems and notation. In Section 32e study well-posedness and existence of stationary solutions for one-component equation of(1.1) for a class of kernels satisfying some growth bounds. More specifically, in Section 3.2we present the main steps of the proofs of a well-posedness result for unbounded coagulationkernels obtained in 1999 by Norris [31] and in Sections 3.3 and 3.4 we review the proofsfor existence and non-existence of stationary solutions to coagulation equation with source,respectively, obtained recently in [13]. In Section 4 we consider the discrete multicomponentequation with constant kernel. Following the computations presented in [21], we computein Section 4.1 explicit time-dependent solutions to the discrete multicomponent coagulationequation with constant kernel K ( x, y ) = 1 and in Section 4.2 stationary solutions when anadditional source at the monomers is present. We also obtain approximations of both solutionsthat show explicitly that mass localizes along a straight line in the multidimensional size spacefor large times and large sizes. Finally in Section 5 we mention some more recent results inthe literature and open questions. By multiplying (1.1) by x and integrating in x from 0 to ∞ one obtains formally an equationfor the mass M ( t ) = (cid:82) ∞ xf ( x, t ) dx given by ddt M ( t ) = 0. This shows that the mass isconserved, provided that the integrals are well-defined. Associated to the mass-conservation,one may write a continuity equation that shows that the mass is transported continuouslyalong the size space: ∂ t ( xf ( x, t )) = ∂ x J ( x, t ) (2.1)where the flux of mass from small to large clusters is given by J ( x, t ) = (cid:90) x (cid:90) ∞ x − y yf ( y, t ) f ( z, t ) K ( y, z ) dzdy (2.2)As we will see in Section 3.3 a (non-equilibrium) stationary solution has a constant flux ofmass at large sizes, i.e., J ( x ) is constant for all x > L , for some positive L . Moreover thisflux plays an important role in the proof of non-existence of stationary solutions in Section3.4.Interestingly, if the coagulation rate is sufficiently large, the mass-conservation is lost.Such phenomenon, called gelation, corresponds to the formation of infinitely large clustersthat are not seen any more by the equations and it may be interpreted as a change in statefrom gas to gel. Mathematically gelation poses interesting challenges [11], however since ithas not been observed in atmospheric aerosols we do not elaborate about it in this text.We note that, contrarily to the Boltzmann equation, the coagulation equation does notpreserve number of particles nor energy. Atmospheric aerosols are suspensions of small particles in the air, whose diameter rangesapproximately between 1 nanometre, in the case of molecular particles, to 100 micrometres,in the case of cloud droplets and dust particles [17]. Aerosols influence sunlight scattering byreflecting and absorbing radiation, and they constitute the seeds that originate the clouds.Therefore, they play an important role in weather and climate forecast [6].3erosols are subject to complex processes that influence their size distribution over time.One important process is the coagulation of particles to produce larger ones. Other processesinclude the formation of new small particles, or monomers, due to certain physical and chem-ical processes, the removal of particles due to gravity or diffusion, and the growth/shrinkagedue to condensation/evaporation [25]. Atmospheric aerosols are typically constituted by dif-ferent chemicals, leading to multicomponent systems, which may alter the rate of the processesmentioned before and consequently, the particle size distribution [35].We consider the regime in which the particles are uniformly distributed in space. Moreover,we assume that removal and growth of particles is not important, which in practice maycorrespond roughly to sizes between 10 nanometers and 10 micrometers [17]. We are then ledto the study of multicomponent systems where particles undergo binary coagulation in thepresence of a source of small particles.Coagulation kernels K have been derived for atmospheric aerosols using kinetic theoryunder several assumptions on the shape and movement of the particles [17]. Aerosol particlesare commonly assumed to be spherical and to undergo elastic collisions with background airparticles. The number of such collision events is assumed to be much larger than the numberof collisions between two coalescing particles, which drives the system towards an equilibriumwhere the particle velocities follow a Maxwell-Boltzmann distribution.Moreover, any collision between coalescing particles yields a coalescing particle. Twodifferent coagulation kernels have been derived under the previous conditions for two differentparticle size regimes, based on the relation between the particle size and the mean free pathin air, ie, the average distance travelled by a particle between two collisions. Under normalpressure and temperature conditions, the mean free path in air, (cid:96) , is of the order of (cid:96) ∼ d p , is much smaller than the meanfree path d p (cid:28) (cid:96) , the particle is more more likely to travel in straight lines before meetinganother coalescing particle. In this case the rate of coagulation has been estimated by thefree molecular coagulation kernel: K α,β = ( α / + β / ) ( α − + β − ) / . (2.3)Otherwise, if the size of a particle is much larger than the mean free path, d p (cid:29) (cid:96) , thecoalescing particle will meet many background air particles before meeting another coalescingparticle. In this case, the air behaves like a fluid and the coalescing particle is more likely todiffuse. The coagulation rate has been estimated by the diffusive coagulation kernel: K α,β = ( α − / + β − / )( α / + β / ) . (2.4)This kernel has been first derived in the original work by Smoluchowski [33]. Other kernelshave been derived under different assumptions on the underlying background gas and particlesfeatures, such as particles moving in a laminar shear or turbulent flow [17], and particleshaving electric charges [34, 35].The behaviour and even the existence of solutions to (1.2) strongly depends on the coag-ulation kernel K α,β . In Sections 3.3 and 3.4 we review existence of stationary solutions fora large class of kernels which includes in particular the free molecular (2.3) and the diffusivekernels (2.4). 4 .3 Derivation of Smoluchowski coagulation equation from particle model The Smoluchowski coagulation equation has been rigorously derived using different approachesthat consider different types of particle systems. In one approach, a purely stochastic particlesystem is considered, where pairs of particles are randomly picked to originate a new particle.The associated stochastic process is usually called Markov-Lushnikov process. A differentapproach considers spatial particle systems, where particles move in an Euclidean domain ina deterministic manner and when they collide they merge with a certain probability.The first approach is inspired in Kac-models for the derivation of the Boltzmann equation[14]. A common strategy is to start from an infinite stochastic particle system where particlesof size x and y coalesce at a rate K ( x, y ) and to prove that the number density, after con-veniently rescaled in such a way that the mean free time is constant, converges, as the unitvolume tends to infinity, to a measure that solves the Smoluchowski coagulation equationwith kernel K . This has been obtained for the additive kernel, product kernel as well as fora class of sub-multiplicative kernels using combinatorial techniques and random graphs. See[2] (Chapter 5.2) for an accessible exposition and [1] for a review on existing results and openproblems.In the second approach, there are fewer rigorous results. The first result to the bestof our knowledge is due to Lang and Xanh [22]. They consider Brownian particles movingin the three-dimensional Euclidean space according to Brownian motion with a diffusioncoefficient D . The particles are assumed to move independently on each other provided theyare at a distance greater than the sum of their radius 2 R . Once they come closer than 2 R they coalesce with probability 1 /
2, forming one Brownian particle with the same radius R and the same diffusion coefficient D . In the limit when the number of particles N goes toinfinity and the radius R goes to zero, such that RN remains constant, the authors provethat the particles remain independent on each other (propagation of chaos) and that thedensity function converges in probability to the solution to the Smoluchowski’s coagulationequation with constant coagulation kernel. The limit where RN remains constant is the so-called Boltzmann-Grad limit and is the limit of constant mean free time. A more generalcase of coalescing Brownian particles with different diffusion coefficients where the diffusioncoefficient changes after coalescence, but not the size R , has been treated in [20]. Morerecently, the change in size after coalescence has been considered in [32] in the case of aone tracer particle moving in a straight line and coalescing with randomly distributed fixedparticles. In this case, a linear coagulation equation with a simple shear kernel was derivedin the kinetic limit where the volume fraction filled by the background of particles tends tozero. We use the notation R ∗ := (0 , ∞ ) and R + := [0 , ∞ ). We denote by M ( I ) the space of signedRadon measures on I ⊂ R + , i.e., the non-negative measures having finite total variation onany compact subset of I , and by (cid:107) · (cid:107) the total variation norm. We denote by M + ( I ) the spaceof measures on M ( I ) that are nonnegative. The measures from M + ( I ) that are also boundedare denoted by M + ,b ( I ) := { µ ∈ M + ( I ) | µ ( I ) < ∞} . The space M + ,b ( I ) equipped with thenorm (cid:107) · (cid:107) is a Banach space. The notation f t ( x ) will sometimes be used to denote f ( t, x ).We denote by C c ( I ) or C b ( I ) the spaces of continuous functions on I that are compactlysupported or bounded, respectively. For simplicity, we use a generic constant C >
We consider a large class of kernels K : (0 , ∞ ) → [0 , ∞ ) satisfying K is continuous , (3.1)and for each ( x, y ) ∈ (0 , ∞ ) , there exist constants c , c , λ and γ such that K ( x, y ) = K ( y, x ) , K ( x, y ) ≥ , (3.2) K ( x, y ) ≥ c [ x λ + γ y − λ + y λ + γ x − λ ] (3.3) K ( x, y ) ≤ c [ x λ + γ y − λ + y λ + γ x − λ ] , (3.4)0 < c ≤ c < ∞ , (3.5)and λ, γ ∈ R . (3.6)Note that condition (3.1) implies that K is measurable . (3.7)This class includes in particular the physical kernels (2.3) and (2.4). The parameter γ represents the homogeneity of the kernel, while λ represents the ”off-diagonal” rate. Theparameter γ yields the behaviour under the scaling of the particle size, while λ measures theimportance of collisions between particles of different sizes. Note that the bounds in (3.3)and (3.4) are homogeneous, i.e., they satisfy for any k > h ( kx, ky ) = k γ h ( x, y ), but thekernels are not necessarily homogeneous.We assume the following conditions on the source η ∈ M + ( R ∗ )supp η ∈ [1 , L ] , for some L > η ( R ∗ ) < ∞ .We consider the following definition of time-dependent solution [31]. Definition 3.1
Assume that K is a measurable function satisfying (3.2) and (3.4) . Let η ≡ . We will say that the map t (cid:55)→ f t : [0 , T ) → M + ( R ∗ ) , where T ∈ (0 , ∞ ] is a localsolution if it satisfies1. for all compact sets B ⊂ R ∗ , t (cid:55)→ f t ( B ) : [0 , T ) → [0 , ∞ ) is measurable2. for all t < T and all compact sets B ⊂ R ∗ (cid:90) t (cid:90) B × R ∗ K ( x, y ) f s ( dx ) f s ( dy ) ds < ∞ , . for all bounded measurable functions ϕ of compact support and t < T it holds (cid:104) ϕ, f t (cid:105) = (cid:104) ϕ, f (cid:105) + (cid:90) t (cid:104) ϕ, L ( f s ) (cid:105) ds (3.9) where L ( f ) is defined by (cid:104) ϕ, L ( f ) (cid:105) = 12 (cid:90) R ∗ (cid:90) R ∗ K ( x, y )[ ϕ ( x + y ) − ϕ ( x ) − ϕ ( y )] f ( dx ) f ( dy ) , (cid:82) R ∗ x x ≤ f ( dx ) < ∞ and (3.9) holds with ϕ ( x ) = x x ≤ .If T = ∞ we call time-dependent solution. One can easily check that condition 2 is the minimal one to have well-defined integrals.Condition 3 is the weak formulation commonly used in the literature and it is obtained byformally multiplying by a test function and integrating in x . Condition 4 is a boundarycondition imposing that no mass enters at 0. Theorem 3.2
Let K a measurable function satisfying (3.2) and (3.4) with λ = 0 and γ < .Let η = 0 . If (cid:104) x , f (cid:105) < ∞ , then there exists a unique time-dependent solution ( f t ) t> . We consider now a source η (cid:54) = 0 of small particles entering into the system at a constantrate and we study the existence of stationary injection solutions, i.e., solutions that satisfy f ( t, x ) = f (0 , x ) for all t > Definition 3.3
Assume that K : R ∗ → R + is a continuous function satisfying (3.2) andthe upper bound (3.4) . Assume further that η ∈ M + ( R ∗ ) satisfies (3.8) . We will say that f ∈ M + ( R ∗ ) , satisfying f ((0 , and (cid:90) R ∗ x γ + λ f ( dx ) + (cid:90) R ∗ x − λ f ( dx ) < ∞ , (3.10) is a stationary injection solution of (1.1) if the following identity holds for any test function ϕ ∈ C c ( R ∗ ) : (cid:90) R ∗ (cid:90) R ∗ K ( x, y ) [ ϕ ( x + y ) − ϕ ( x ) − ϕ ( y )] f ( dx ) f ( dy ) + (cid:90) R ∗ ϕ ( x ) η ( dx ) = 0 . (3.11)Condition (3.10) is the minimal one for the integrals in (3.11) be well-defined. Stationaryinjection solutions have a constant in time flux of mass from small to large sizes, due to thesource, therefore they are non-equilibrium solutions. Note that to be able to be stationary,the volume of particles entering the system has to balance the volume of particles leaving thesystem. Interestingly, there is an implicit removal of particles from the system at infinite sizesthat allows the existence of these solutions. As we see in the next two Theorems, for someclass of coagulation rates, including the diffusive kernel (2.4), such balance exists, while forother class of kernels, including the free molecular kernel (2.3), such balance does not exist. Theorem 3.4
Assume that K satisfies (3.2) – (3.6) and | γ + 2 λ | < . Let η (cid:54) = 0 satisfy (3.8) .Then, there exists a stationary injection solution f ∈ M + ( R ∗ ) , f (cid:54) = 0 , to (1.1) in the senseof Definition 3.3. heorem 3.5 Suppose that K ( x, y ) satisfies (3.2) – (3.6) as well as | γ + 2 λ | ≥ . Let usassume also that η (cid:54) = 0 satisfies (3.8) . Then, there is not any solution of (1.1) in the senseof the Definition 3.3. Note that the diffusive kernel (2.4) satisfies the growth conditions (3.3)-(3.4) with γ = 0and λ = 1 /
3, while the free molecular kernel (2.3) satisfies the growth conditions with γ = 1 / λ = 1 /
2. Therefore there exists a stationary solution for the diffusive but not for the freemolecular kernel.The mass flux from small to large sizes associated to a stationary injection solutions isgiven in the next Lemma.
Lemma 3.6
Suppose that the assumptions of Theorem 3.4 hold. Let f be a stationary injec-tion solution in the sense of Definition 3.3. Then f satisfies for any R > J ( R ) = (cid:90) (0 ,R ] xη ( dx ) (3.12) with J ( R ) = (cid:90) (0 ,R ] (cid:90) ( R − x, ∞ ) K ( x, y ) xf ( dx ) f ( dy ) Remark 3.7 If R ≥ L η , the right-hand side of (3.12) is constant equal to J η = (cid:82) [1 ,L η ] xη ( dx ) >
0. Therefore, J ( R ) = J η for R > L η , i.e., the mass flux is constant in the regions that includelarge sizes. Proof: [Idea of the proof] For for each ε >
0, define a test function ϕ ( x ) = xχ ε ( x ) ∈ C c ( R ∗ )where χ ε ∈ C ∞ c ( R ∗ ) is such that 0 ≤ χ ε ≤ χ ε ( x ) = 1, for 1 ≤ x ≤ R , and χ ε ( x ) = 0, for x ≥ R + ε . Then using the test function in (3.11) the result follows after some computationsand after letting ε → (cid:3) We describe the main ideas of the proofs obtained in [31] (Section 2).The first step is to prove well-posedness for a truncated problem. The second step is tocompute estimates that allow us to remove the truncation and to obtain well-posedness forthe original problem.Let B ⊂ R ∗ be a compact set. We consider the space M ( B ) of finite signed measuressupported on B . Note that all measures in M ( B ) are bounded. Note that from the hypothesesof Theorem 3.2 on the kernel we have that0 ≤ K ( x, y ) ≤ w ( x ) + w ( y ) , with w ( x ) := x γ and γ < . (3.13)The truncated operator L B : M ( B ) × R → M ( B ) × R is defined by (cid:104) ( ϕ, a ) , L B ( f, λ ) (cid:105) :=12 (cid:90) R ∗ × R ∗ { ϕ ( x + y ) { x + y ∈ B } + aw ( x + y ) { x + y / ∈ B } − ϕ ( x ) − ϕ ( y ) } × K ( x, y ) f ( dx ) f ( dy )+ λ (cid:90) R ∗ { aw ( x ) − ϕ ( x ) } w ( x ) f ( dx )8or all bounded measurable functions ϕ on R ∗ and all a ∈ R , where (cid:104) ( ϕ, a ) , ( f, λ ) (cid:105) denotes (cid:104) ϕ, f (cid:105) + aλ . The truncated equation reads (cid:104) ( ϕ, a ) , ( f t , λ t ) (cid:105) = (cid:104) ( ϕ, a ) , ( f , λ ) (cid:105) + (cid:90) t (cid:104) ( ϕ, a ) , L B ( f, λ ) (cid:105) ds. (3.14)An interpretation of the operator L B is the following (see [31] for more details). Particlesof size x and y merge at a rate K ( x, y ) and they produce a new particle of size x + y . If themerging particle has size outside B , we add w ( x + y ) to λ .A solution to (3.14) is defined next. Definition 3.8
Let T ∈ (0 , ∞ ) . We will say that ( f t , λ t ) t ∈ [0 ,T ] is a local solution to (3.14) if t (cid:55)→ ( f t , λ t ) : [0 , T ] → M ( B ) × R is a continuous map satisfying (3.14) for all t ∈ [0 , T ] .Additionally, ( f t ) t ∈ [0 ,T ] is called a solution when [0 , T ] is replaced by [0 , ∞ ) . Proposition 3.9
Suppose that f ∈ M ( B ) with f ≥ and λ ∈ [0 , ∞ ) . The equation (3.14) has a unique solution ( f t , λ t ) t ≥ starting from ( f , λ ) . Moreover f t ≥ and λ t ≥ for all t ≥ .Proof: [idea of the proof] The first step is to show that there is a constant T > γ and B such that there exists a unique local solution ( f t , λ t ) t ∈ [0 ,T ] to (3.14) startingfrom ( f , λ ). This is obtained by using an iterative scheme of continuous maps ( f nt , λ nt ) :[0 , ∞ ) (cid:55)→ M ( B ) × R defined by( f t , λ t ) = ( f , λ )( f nt , λ nt ) = ( f , λ ) + (cid:90) t L B ( f n − t , λ n − t )and proving that there exists a T > f n , λ n ) converges in M ( B ) × R uniformlyin t ≤ T to the desired local solution, which is also unique.The second step is to prove that f t ≥ , t ∈ [0 , T ], which is obtained using again aniterative argument similar to the one used in the first step.Finally, the third step is to show that the solution exists for all times t ∈ [0 , ∞ ). Choosing ϕ = w and a = 1 we obtain that ddt ( (cid:104) w, f t (cid:105) + λ t ) = 12 (cid:90) R ∗ × R ∗ { w ( x + y ) − w ( x ) − w ( y ) } K ( x, y ) f t ( dx ) f t ( dy ) ≤ , which implies that (cid:107) f T (cid:107) + | λ T | ≤ (cid:104) w, f T (cid:105) + λ T ≤ (cid:104) w, f (cid:105) + λ , using a scaling argument, we may assume without loss of generality that (cid:104) w, f (cid:105) + λ ≤ (cid:107) f T (cid:107) + | λ T | ≤
1. We can start again from ( f T , λ T ) at time T to extend thesolution to [0 , T ] and so on. Moreover, choosing ϕ = 0 and a = 1 in (3.14), we obtain ddt λ t = 12 (cid:90) R ∗ × R ∗ { w ( x + y ) ¯ B ( x + y ) } K ( x, y ) f ( dx ) f ( dy ) + λ t (cid:90) R ∗ w ( x ) f ( dx ) , which implies that λ t ≥
0, for all t ≥
0, due to f t ≥
0, which ends the proof of the proposition. (cid:3) roof: [Theorem 3.2] Fix f ∈ M + , such that (cid:104) w, f (cid:105) < ∞ . For each compact set B ⊂ R ∗ define f B = B f and λ B = (cid:82) ¯ B w ( x ) f ( dx ). From Proposition 3.9 there is a unique solution( f B , λ Bt ) t ≥ to (3.14) starting from ( f B , λ B ). We now set f t = lim B → R ∗ f Bt and λ t = lim B → R ∗ λ Bt .Using (3.13), we obtain by dominated convergence, ddt (cid:104) ϕ, f t (cid:105) = 12 (cid:90) R ∗ × R ∗ { ϕ ( x + y ) − ϕ ( x ) − ϕ ( y ) } K ( x, y ) f t ( dx ) f t ( dy ) − λ t (cid:104) ϕw, f t (cid:105) , for all bounded measurable functions ϕ . One can prove that for all t < T and for any localsolution ( g t ) t
0, which allows to pass to the limit as B → R ∗ in (3.14)and to deduce that λ t = 0 , t > . (3.16)Then (3.15) and (3.16) imply that ( f t ) t ≥ is a solution and moreover, it is the only solution. (cid:3) We present the main ideas of the proofs obtained in [13].The general strategy to prove existence of stationary solutions is similar to the one usedin the proof of well-posedness presented in the previous Section. First, we prove existence ofa stationary solution for a truncated problem and second, we obtain estimates that allow toremove the truncation and to obtain the existence result for the original problem. Unfortu-nately the method used to prove existence does not give uniqueness, that problem needs aseparate treatment (see [23] for a simple explanation of the available techniques).Let ε > R ∗ ≥ L η , where L η is the upper bound of the support of the source η defined in (3.8). We will eventually make ε → R ∗ → ∞ . We consider kernels K ε,R ∗ that are continuous, bounded and have compact support, such that K ε,R ∗ ( x, y ) ≤ a ( ε ) , ( x, y ) ∈ R (3.17) K ε,R ∗ ( x, y ) ∈ [ a ( ε ) , a ( ε )] , ( x, y ) ∈ [1 , R ∗ ] (3.18) K ε,R ∗ ( x, y ) = 0 , x ≥ R ∗ or y ≥ R ∗ , (3.19)and lim R ∗ →∞ K ε,R ∗ ( x, y ) = K ε ( x, y ) (3.20)where K ε is continuous and satisfies K ε ( x, y ) ∈ [ a ( ε ) , a ( ε )] , for all ( x, y ) ∈ R andlim ε → K ε ( x, y ) = K ( x, y ) . (3.21)Additionally, in the evolution equation, we consider a cut-off of the gain term due to thecoagulation that ensures that the measure solutions are supported in [1 , R ∗ ] and boundedat all times. To this end, we choose ζ R ∗ ∈ C ( R ∗ ) such that 0 ≤ ζ R ∗ ≤ ζ R ∗ ( x ) = 1 for10 ≤ x ≤ R ∗ , and ζ R ∗ ( x ) = 0 for x ≥ R ∗ . The regularized time evolution equation then readsas ∂ t f ( x, t ) = ζ R ∗ ( x )2 (cid:90) (0 ,x ] K ε,R ∗ ( x − y, y ) f ( x − y, t ) f ( y, t ) dy − (cid:90) R ∗ K ε,R ∗ ( x, y ) f ( x, t ) f ( y, t ) dy + η ( x ) . (3.22) Definition 3.10
Let ε > and R ∗ ≥ L η . Suppose that K ε,R ∗ satisfies (3.17) - (3.19) and η ∈ M + ( R + ) satisfies (3.8) . Consider some initial data f ∈ M + ( R ∗ ) for which f ((0 , ∪ (2 R ∗ , ∞ )) =0 . Then f ∈ M + ,b ( R ∗ ) . We will say that f ∈ C ([0 , T ] , M + ,b ( R ∗ )) satisfying f ( · ,
0) = f ( · ) is a time-dependent solution of (3.22) if the following identity holds for any test function ϕ ∈ C ([0 , T ] , C c ( R ∗ )) and all < t < T , ddt (cid:90) R ∗ ϕ ( x, t ) f ( dx, t ) − (cid:90) R ∗ ˙ ϕ ( x, t ) f ( dx, t )= 12 (cid:90) R ∗ (cid:90) R ∗ K ε,R ∗ ( x, y ) [ ϕ ( x + y, t ) ζ R ∗ ( x + y ) − ϕ ( x, t ) − ϕ ( y, t )] f ( dx, t ) f ( dy, t )+ (cid:90) R ∗ ϕ ( x, t ) η ( dx ) , (3.23) where ˙ ϕ denotes the Fr´echet time-derivative of ϕ . Proposition 3.11
Let ε > and R ∗ ≥ L η . Suppose that K ε,R ∗ satisfies (3.17) - (3.19) and η ∈ M + ( R + ) satisfies (3.8) . Then, for any initial condition f satisfying f ∈ M + ( R ∗ ) , f ((0 , ∪ (2 R ∗ , ∞ )) = 0 there exists a unique time-dependent solution f ∈ C ([0 , T ] , M + ,b ( R ∗ )) to (3.22) which solves it in the classical sense. Moreover, f is a Weak solution of (3.22) inthe sense of Definition 3.10 such that f ((0 , ∪ (2 R ∗ , ∞ ) , t ) = 0 , for ≤ t ≤ T , and the following estimate holds (cid:90) R ∗ f ( dx, t ) ≤ (cid:90) R ∗ f ( dx ) + Ct , t ≥ , for C = (cid:82) R ∗ η ( dx ) ≥ which is independent of f , t , and T .Proof: [Idea of the proof] Since the kernel is bounded, the result may be obtained using Banachfixed-point theorem. (cid:3) Definition 3.12
Let ε > and R ∗ ≥ L η . Suppose that K ε,R ∗ satisfies (3.17) - (3.19) and η ∈ M + ( R + ) satisfies (3.8) . We will say that f ∈ M + ( R ∗ ) , satisfying f ((0 , ∪ (2 R ∗ , ∞ )) = 0 is a stationary injection solution of (3.22) if the following identity holds for any test function ϕ ∈ C c ( R ∗ ) : (cid:90) R ∗ (cid:90) R ∗ K ε,R ∗ ( x, y ) [ ϕ ( x + y ) ζ R ∗ ( x + y ) − ϕ ( x ) − ϕ ( y )] f ( dx ) f ( dy )+ (cid:90) R ∗ ϕ ( x ) η ( dx ) .
11e denote by S ( t ) the semigroup defined by the time-dependent solution f obtained inProposition 3.11, S ( t ) f = f ( · , t )that satisfies the semigroup property S ( t + s ) f = S ( t ) S ( s ) f, t, s ∈ R + . The operators S ( t ) define mappings S ( t ) : X R ∗ → X R ∗ , quad for each t , t ∈ R + with X R ∗ = { f ∈ M + ( R ∗ ) : f ((0 , ∪ (2 R ∗ , ∞ )) = 0 } . Proposition 3.13
Under the assumptions of Proposition 3.11, there exists a stationary in-jection solution ˆ f ∈ M + ( R ∗ ) to (3.22) as defined in Definition 3.12.Proof: [Idea of the proof] The key point of the proof is to use Schauder fixed point theorem.The first step is to obtain the existence of an invariant region for the evolution problem (3.23).To that end, we choose a time independent test function ϕ ( x ) = 1 for x ∈ [1 , R ∗ ]. Usingthe lower bound for the kernel (3.18) and that f ( · , t ) has support in [1 , R ∗ ] we obtain thefollowing estimate ddt (cid:90) [1 , R ∗ ] f ( dx, t ) ≤ − a (cid:32)(cid:90) [1 , R ∗ ] f ( dx, t ) (cid:33) + c where c = (cid:82) R ∗ η ( dx ). This implies that for a large enough M >
0, the set U M = (cid:40) f ∈ X R ∗ : (cid:90) [1 , R ∗ ] f ( dx ) ≤ M (cid:41) . is invariant under the time evolution (3.22). Moreover, U M is compact in the ∗− weak topologydue to Banach-Alaoglu’s Theorem (cf.[3]), since it is an intersection of a ∗− weak closed set X R ∗ and the closed ball (cid:107) f (cid:107) ≤ M .The second step is to prove that for each t >
0, both maps S ( t ) : U M → U M and t (cid:55)→ S ( t ) f are continuous in the ∗− weak topology.Finally, the third step of the proof reads as follows. Since for each t , the operator S ( t )is continuous and U M is compact and convex when endowed with the ∗− weak topology, wecan apply Schauder fixed point theorem to conclude that for all δ > f δ of S ( δ ) in U M . Moreover, since U M is metrizable and hence sequentially compact, thereis a convergent sequence { f δ n } n ∈ N , i.e., there exists ˆ f ∈ U M such that f δ n → ˆ f when δ n → ∗− weak topology. For each t we choose δ n = t/n . Using the semigroup property weobtain that S ( t ) f δ n = S ( nδ n ) f δ n = S ( δ n ) f δ n . Using the continuity of t (cid:55)→ S ( t ) f and the factthat S (0) ˆ f = ˆ f , we obtain S ( δ n ) f δ n → ˆ f . On the other hand using the continuity of S ( t )we obtain that S ( t ) f δ n → S ( t ) ˆ f . Therefore S ( t ) ˆ f = ˆ f and thus ˆ f is a stationary solution to(3.22), which concludes the proof. (cid:3) The next Lemma provides uniform estimates for integrals.12 emma 3.14
Let a > , R ≥ a and b ∈ (0 , be such that bR > a . Suppose f ∈ M + ( R ∗ ) , ϕ ∈ C ( R ∗ ) , g ∈ L ( R ∗ ) , and g, ϕ ≥ . If z (cid:90) [ bz,z ] ϕ ( x ) f ( dx ) ≤ g ( z ) , for z ∈ [ a, R ] , then (cid:90) [ a,R ] ϕ ( x ) f ( dx ) ≤ (cid:82) [ a, ∞ ) g ( z ) dz ln( b − ) + Rg ( R ) . We now extend the previous existence result to general unbounded kernels K supportedin R and satisfying the conditions of the theorem 3.4. Proof: [idea of the proof of Theorem 3.4] Let f ε,R ∗ be a stationary injection solution to (3.22)as defined in Definition 3.12 provided by Proposition 3.13. The idea is to obtain estimatesthat are independent on both ε and R ∗ that allow to pass to the limit as ε → R ∗ → ∞ and to obtain the existence of a stationary injection solution to the original problem as definedin (3.3).First we obtain an estimate uniform in R ∗ : (cid:90) [0 , R ∗ / f ε,R ∗ ( dx ) ≤ ¯ C ε , R ∗ > , where ¯ C ε is a constant independent on R ∗ . This estimate implies, that taking a subsequenceif needed, there exists f ε ∈ M + ( R + ) such that f ε ([0 , f ε,R n ∗ (cid:42) f ε as n → ∞ in the ∗ − weak topologywith R n ∗ → ∞ as n → ∞ . For any bounded continuous test function ϕ : [0 , ∞ ) → R , oneproves that f ε satisfies12 (cid:90) [0 , ∞ ) K ε ( x, y ) [ ϕ ( x + y ) − ϕ ( x ) − ϕ ( y )] f ε ( dx ) f ε ( dy ) + (cid:90) [0 , ∞ ) ϕ ( x ) η ( dx ) = 0 . where K ε is defined in (3.20).Second, we obtain estimates independent on ε :1 z (cid:90) [ z ,z ] f ε ( dx ) ≤ ˜ Cz (cid:32) (cid:0) z γ , ε (cid:1) (cid:33) , (3.24)and 1 z (cid:90) [ z ,z ] f ε ( dx ) ≤ ˜ Cz √ ε . where ˜ C is independent on ε . This estimate yields ∗− weak compactness of the family ofmeasures { f ε } ε> in M + ( R + ) . Therefore, there exists f ∈ M + ( R + ) such that: f ε n (cid:42) f as n → ∞ in the ∗ − weak topologyfor some subsequence { ε n } n ∈ N with lim n →∞ ε n = 0 . Using (3.21) and Lemma 3.14, one canprove that f ε satisfies (3.11) for any ϕ ∈ C c ( R + ). In particular, f (cid:54) = 0 due to η (cid:54) = 0.13t only remains to prove (3.10). Taking the limit of (3.24) as ε → z (cid:90) [2 z/ ,z ] f ( dx ) ≤ (cid:101) Cz / γ/ for all z ∈ (0 , ∞ ) , which implies 1 z (cid:90) [2 z/ ,z ] x µ f ( dx ) ≤ (cid:101) C z µ z / γ/ for all z ∈ (0 , ∞ ) , for any µ ∈ R . From Lemma 3.14 we obtain the boundedness of the moment of order µ : (cid:90) [0 , ∞ ) x µ f ( dx ) < ∞ . for any µ satisfying µ < γ +12 . In particular, since | γ + 2 λ | <
1, then the moments µ = − λ and µ = γ + λ are bounded, which proves (3.10). (cid:3) We present the main ideas of the proof obtained in [13].The proof is done by contradiction. Let the kernel K satisfy the power law bounds (3.3)-(3.4) with | γ + 2 λ | ≥
1. Suppose that f ∈ M + ( R ∗ ) is a stationary injection solution of (1.1)in the sense of Definition 3.3. Then f satisfies the weak formulation (3.11) as well as thecondition on the moments (3.10).The first step is to rewrite (3.11) using the flux formulation. Consider the function J : R ∗ → R + defined by J ( R ) = (cid:90) (cid:90) Σ R K ( x, y ) xf ( dx ) f ( dy ) (3.25)where Σ R = { x ≥ , y ≥ x + y > R, x ≤ R } . Let ε > , R ≥ χ ε ∈ C ∞ ( R + ) satisfy χ ε ( x ) = 1 , x ≤ R and χ ε ( x ) = 0 , x ≥ R + ε .Choosing a test function ϕ ( x ) = xχ ε ( x ) we obtain from (3.11) the flux formulation J ( R ) = (cid:90) [1 ,R ] xη ( dx ) , R ≥ . We note that J describes the flux of particles passing through Σ R and that this flux is constantfor all R ≥ L η and equal to J ( L η ) = (cid:82) [1 , ∞ ) xη ( dx ) >
0, i.e., J ( R ) = J ( L η ) , R ≥ L η . The second step is to prove that the main contribution to the integral (3.25) as R → ∞ isdue to collisions between particles of size close to R and particles of size of order 1. To thatend, for a given δ > R = D (1) δ ∩ D (2) δ such that D (1) δ = { x ≥ , y ≥ y ≤ δx } ,D (2) δ = { x ≥ , y ≥ y > δx } J k ( R ) = (cid:90) (cid:90) Σ R ∩ D kδ K ( x, y ) xf ( dx ) f ( dy ) , k = 1 , . Therefore J ( R ) = J ( R ) + J ( R ) . Using the upper bound for the kernel (3.4), the moment condition (3.10) and the fact thatΣ R ∩ D δ ⊂ [1 , R ] × [ δR δ , ∞ ) one concludes after some computations that the contribution of J vanishes as R → ∞ , i.e., lim R →∞ J ( R ) = 0which implies that lim R →∞ J ( R ) = lim R →∞ J ( R ) = J ( L η ) . In the remainder of the proof we will use the notation a := γ + λ and b := − λ if ( γ +2 λ ) ≥ a := γ + λ and b := − λ if ( γ + 2 λ ) ≤ −
1. Then, the assumption (3.10) may be rewritten as (cid:90) R ∗ x a f ( dx ) < ∞ . (3.26)The third step of the proof consists in obtaining a lower bound for the fluxes that implies alower bound for the number of particles in some region of the size space. Using the upperbound for the kernel (3.4) we obtain after some computationslim inf R →∞ (cid:32) R a +1 (cid:90) (cid:90) Σ R ∩ D (1) δ y b f ( dx ) f ( dy ) (cid:33) ≥ J ( L η ) c (cid:0) δ | a − b | (cid:1) . (3.27)For R sufficiently large we have thatΣ R ∩ D (1) δ ⊂ { ( x, y ) : 1 ≤ y ≤ δR, R < x + y, x ≤ R } whence, (3.27) implies the inequality (cid:90) [1 ,δR ] y b f ( dy ) (cid:90) ( R − y,R ] f ( dx ) ≥ J ( L η )2 c (cid:0) δ | a − b | (cid:1) R a +1 (3.28)for R ≥ R with R large enough. We now consider two cases separately a ≥ a < a ≥
0. Due to (3.26) we may define F ( R ) = (cid:90) ( R, ∞ ) f ( dx ) , R ≥ . (3.29)Using (3.29) we can rewrite (3.28) as − (cid:90) [1 ,δR ] [ F ( R − y ) − F ( R )] y b f ( dy ) ≤ − J ( L η )2 c (cid:0) δ | a − b | (cid:1) R a +1 for R ≥ R . Then, using a comparison argument (see Lemma 4.1 in [13]), for some constant
B >
0, itfollows that F ( R ) ≥ BR a if R ≥ R , for a > , (3.30)15nd F ( R ) ≥ B log( R ) if R ≥ R , for a = 0 . (3.31)In the case a >
0, (3.30) implies B ≤ R a F ( R ) ≤ (cid:90) ( R, ∞ ) x a f ( dx )Taking the limit when R → ∞ and using (3.26) it follows that B ≤
0, which leads to acontradiction. In the case a = 0, the contradiction follows from (3.31) in a similar way using(3.26).Let now a <
0. We define the function F by F ( R ) = (cid:90) [1 ,R ] f ( dx ) , R ≥ . (3.32)Using (3.32) we can rewrite (3.28) as: − (cid:90) [1 ,δR ] [ F ( R ) − F ( R − y )] y b f ( dy ) ≤ − J ( L η )2 c (cid:0) δ | a − b | (cid:1) R a +1 for R ≥ R . As in the previous case, using a comparison argument (see Lemma 4.2 in [13]), it follows thatthere is
B > F ( R ) ≥ BR a if R ≥ R . For a small ε > ε < B there exists M such that (cid:90) [ M, ∞ ) x a f ( dx ) = ε . Then for all
R > M we have B ≤ R a (cid:90) [1 ,R ] f ( dx ) ≤ R a (cid:90) [1 ,M ] f ( dx ) + (cid:90) [ M,R ] x a f ( dx ) ≤ R a (cid:90) [1 ,M ] f ( dx ) + ε. Since a <
0, taking the limit as R → ∞ we obtain B ≤ ε , which leads to a contradiction. We consider the multicomponent discrete equation with constant kernel K α,β = 2, dn α ( t ) dt ( t ) = (cid:88) β<α n β ( t ) n α − β ( t ) − n α ( t ) (cid:88) β> n β ( t ) (4.1)with initial condition n α (0) = 1 d (cid:88) | β | =1 δ α,β (4.2)16here β > α denotes β i ≥ α i for all i = 1 , ..., d and β > α , and δ α,β = 1 if α = β and δ α,β = 0otherwise. Note that the initial condition (4.2) is supported in the monomers and its massis uniformly distributed by the types of particles. The initial mass of each type of particle is d . Existence and uniqueness of a solution to (4.1)-(4.2) in the onecomponent case d = 1 isproven in [30] for any initial condition satisfying (cid:80) ∞ α =1 n α (0) < ∞ using Laplace transforms.An explicit solution to the multicomponent problem has been obtained in [27] and [21] usinga generating function.In this Section we review the computations described in [21] to obtain an explicit solutionto (4.19). We then study the long-time behaviour using an approximation of the solution forlarge times and large sizes obtained in [21], in particular, we observe the phenomenon of masslocalization along a straight line in the size space.Multiplying (4.19) by a compactly supported test function ψ α and summing in α we obtainthe weak formulation ∂ t ∞ (cid:88) α =1 ψ α n α ( t ) = ∞ (cid:88) α,β =1 [ ψ α + β − ψ α − ψ β ] n α ( t ) n β ( t ) (4.3)The solution may be obtained using the generating function F : R d × R + → R defined by F ( z, t ) = (cid:88) α> z α n α ( t ) (4.4)where z α = z α z α ...z α d d . Using ψ α = z α in (4.3) we obtain an equation for F , ∂ t F ( z, t ) = F ( z, t ) − F ( z, t ) N ( t ) (4.5)where N ( t ) = F (0 , t ) = (cid:80) α> n α is the total number of particles at time t . From (4.2) theinitial number of particles is N (0) = 1. Using ψ α = 1 in (4.3) we obtain an equation for N , ∂ t N ( t ) = − N ( t ) , N (0) = 1 ⇐⇒ N ( t ) = 11 + t (4.6)If we subtract equations (4.5) and (4.6) we obtain an equation for F − N , ∂ t ( F − N ) = ( F − N ) .Solving this equation and using (4.6) yields an expression for FF ( z, t ) = F ( z )(1 + t )(1 + t − tF ( z )) (4.7)where F ( z ) = F ( z,
0) is given by F ( z ) = d (cid:80) di =1 z i after substituting (4.2) in (4.4). Theexpression for F will be used in the following to determine the solution to (4.1).We note that if { n α } α> is a solution to the multicomponent coagulation equation (4.1),then { n | α | } α> , where | α | = (cid:80) di =1 α i is the sum variable and n | α | is defined by n | α | = (cid:80) β> n β δ | α | , | β | , is a solution to the one component equation with constant kernel K | α | , | β | = 2and initial condition n | α | (0) = δ | α | , . This result may be obtained using the weak formulation(4.3) with a test function of the form ψ α = ϕ | α | . We first solve the one component equationto find an expression for { n | α | } α> .We consider the generating function f : R × R + → R associated to the one componentproblem f ( z, t ) = ∞ (cid:88) | α | =1 z | α | n | α | ( t ) , (4.8)17hich may be expressed by (4.7) with f ( z, t ) = (cid:80) ∞| α | =1 z | α | n | α | (0) = z , i.e. f ( z, t ) = z (1 + t )(1 + t − tz ) (4.9)Using the Taylor series, we expand f around z = 0 and obtain f ( z, t ) = ∞ (cid:88) k =1 z k t k − (1 + t ) k +1 , z < tt , t > . (4.10)Comparing each term of the two series (4.10) and (4.8) we conclude that the solution to theone component equation is n | α | ( t ) = t | α |− (1 + t ) | α | +1 . (4.11)The solution to the multicomponent equation (4.1) can now be computed by expanding(4.7) and comparing with (4.4). Using the Taylor series in several variables we obtain theexpansion of (4.7) around 0, f ( z, t ) = ∞ (cid:88) k =1 d k t k − (1 + t ) k +1 | z | k , | z | < d tt , t > . (4.12)Comparing with (4.8) and using (4.11) and the fact that ( z + ... + z d ) k = (cid:80) | α | = k k ! α ! α ! ...α d ! z α z α ...z α d d we finally obtain the solution to the multicomponent coagulation equation (4.1) expressed interms of n | α | , n α ( t ) = n | α | ( t ) g ( α ) with g ( α ) = 1 d k k ! α ! α ! ...α d ! . (4.13)To study the long time behaviour, we use the fact that lim( tt ) t = e to obtain an approx-imation for n | α | ( t ) for large | α | and large time t such that | α | ∼ tn | α | ( t ) ≈ t − exp( − | α | t ) , t > Remark 4.1
In [30] it was proven that n ( x, t ) = t − exp (cid:0) − xt (cid:1) is in fact the limit of arescaled solution, provided the initial mass is either finite, which includes the case treated inthis Section, or its distribution function diverges sufficiently weakly.We also consider an approximation of the function gg ( α ) ≈ | α | − ( d − / exp( − | α | − | α | ) (4.15)where | α | − = d d (cid:80) i,j =1 ( α i − α j ) denotes the generalized mass difference variable. Using (4.14)and (4.15) in (4.13) we obtain for large t and | α | the approximation n α ( t ) ≈ t − | α | − ( d − / exp( − | α | t ) exp( − | α | − | α | ) . (4.16)18e observe that besides the mass scale imported from the solution to the one componentequation, there is a second mass scale given by | α | − ∼ √ t . Introducing the variables ξ = | α | t and ρ = | α | − √ t we may then write the solution in a scaling form n α ( t ) ≈ t − ( d +3) / φ ( ξ, ρ ) (4.17)where φ ( ξ, ρ ) = ξ − ( d − / exp( − ξ ) exp( − ρ ξ ) (4.18)Finally we note from (4.16) that for any fixed time, n α ( t ) reaches maximum values when | α | − = 0. This condition defines a straight line in the size space given by { α ∈ N d + | α = α = ... = α d } . Mass localization is also observed in stationary solutions to coagulation equations with sourceby applying a similar study as in the previous Section. We consider the stationary multicom-ponent coagulation equation with source and constant kernel as before K α,β = 2,0 = (cid:88) β<α n α − β n β − (cid:88) β> n α n β + s α (4.19)where s α is the source term. In analogy to the initial conditions in the time-dependent case(4.2), the source term is given by s α = hd (cid:88) | β | =1 δ α,β , (4.20)for some given h > ψ α , the weak formulation is now given by0 = ∞ (cid:88) α,β =1 [ ψ α + β − ψ α − ψ β ] n α n β + hd (cid:88) | α | =1 ψ α . (4.21)The generating function F ( z ) = ∞ (cid:88) | α | =1 z | α | n α (4.22)satisfies F ( z ) − F ( z ) N + S ( z ) = 0 (4.23)where S ( z ) = hd (cid:80) | α | =1 z α = hd (cid:80) di =1 z i and N = √ h is obtained using an appropriate testfunction in (4.21). The solution to (4.23) reads F ( z ) = √ h [1 − (cid:114) − | z | d ] . (4.24)19igure 1: Representation of an approximation of a stationary solution to the two componentcoagulation equation with source and constant kernel (4.19). We observe a concentration ofparticles along a straight line.The solution to (4.19) is obtained by expanding F in powers of the variables z i and comparingwith (4.22), yielding n α = n | α | g ( α ) (4.25)where g is defined in (4.13) and n | α | = √ h (2 | α | )!(2 | α | − | α | | α | !) . (4.26)For large sizes we may approximate n | α | by √ h | α | − / , therefore using also (4.15), we obtain n α ≈ √ h | α | − ( d +2) / exp( − | α | − | α | ) . (4.27)Like in the time-dependent problem, an additional size scale is observed | α | − ∼ (cid:112) | α | . Alsohere we can see from (4.27) that a stationary solution n α reaches maximum values at thestraight line defined by { α ∈ N d + | α = α = ... = α d } . A representation of (4.27) is shownin Figure 4.2. The existence and uniqueness result of a time-dependent solution have also been established[31] for coagulation kernels satisfying (3.3)-(3.4) with γ + λ = − λ and λ > − / f is a function, for a class of kernels satisfying (3.3)-(3.4) with c = c = 1, λ ∈ [ − , γ ∈ [0 , γ ≤ − λ , γ + λ ∈ [ − ,
1] and ( γ, λ ) (cid:54) = ( − λ, − γ ≤ λ = 0, and for a different class of kernels with γ ∈ (1 , λ = 1 up to a gelation time T .20n general well-posedness remains an open problem. We refer to the survey [24] for furtherreferences.In the presence of a constant source of monomers, the existence and non-existence resultspresented in Section 3 are the most recent ones to the best of our knowledge. Previous results[8] have been obtained for particular classes of kernels that are included in the more generalsetting presented here. In the multicomponent equation with kernel K satisfying c w ( x, y ) ≤ K ( x, y ) ≤ c w ( x, y ) with w ( x, y ) = (cid:88) i x γ i − λ i y λ i + y γ i − λ i x λ i , (5.1)we expect the existence and nonexistence results to be still valid for each class of kernelssatisfying | γ i + 2 λ i | < i and | γ i + 2 λ i | ≥ i , respectively. Similarly to thetime-dependent solutions, we also expect that stationary solutions exhibit localization alonga straight line provided the bounds for the kernel are invariant under permutations.Almost nothing is known about rigorous results for the multicomponent coagulation equa-tions with general kernels. However, the well-posedness results for the one-component case,are expected to also hold in the multicomponent case provided the kernel satisfies the bounds(5.1) with γ i and λ i satisfy the same conditions for well-posedness in dimension d = 1 forall i . Mass localization for large times is expected to hold for a class of kernels that satisfythe same bounds with the additional condition that w is invariant under any permutation ofthe components. An additive kernel that does not satisfy this invariance has been consideredin [36]. We expect that in that case the mass does not localize in a straight line, becauseeach component will have a different rate of coagulation and complex multiscale behaviour isexpected to emerge that may break down the nice localization structure.Mass localization results are very important in the optimization of current algorithms asthey allow to focus the computations on the region of size space where the mass is expectedto be localized. The computational complexity of the multicomponent problem may in thisway be reduced to the complexity of the onecomponent problem.There are also many open problems for more general coagulation equations with fragmen-tation and additional sink and growth terms as well as on the derivation of these equationsfrom appropriate particle systems. We refer to [7] for a brief overview on some of these topics. Acknowledgements.
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M. A. Ferreira
Department of Mathematics and Statistics, University of Helsinki,P.O. Box 68, FI-00014 Helsingin yliopisto, FinlandE-mail: [email protected]@helsinki.fi