Uniqueness of post-gelation solutions of a class of coagulation equations
UUNIQUENESS OF POST-GELATION SOLUTIONS OF A CLASSOF COAGULATION EQUATIONS
RAOUL NORMAND AND LORENZO ZAMBOTTI
Abstract.
We prove well-posedness of global solutions for a class of coagulationequations which exhibit the gelation phase transition. To this end, we solve anassociated partial differential equation involving the generating functions beforeand after the phase transition. Applications include the classical Smoluchowskiand Flory equations with multiplicative coagulation rate and the recently intro-duced symmetric model with limited aggregations. For the latter, we compute thelimiting concentrations and we relate them to random graph models. Introduction
Coagulation models.
In this paper we deal with the problem of uniquenessof post-gelation solutions of several models of coagulation, namely Smoluchowski’sand Flory’s classical models, and the corresponding models with limited aggregationsrecently introduced by Bertoin [3].Smoluchowski’s coagulation equations describe the evolution of the concentrationsof particles in a system where particles can perform pairwise coalescence, see e.g.[1, 18, 23]. In the original model of Smoluchowski [29], a pair of particles of mass,respectively, m and m (cid:48) , coalesce at rate κ ( m, m (cid:48) ) and produce a particle of mass m + m (cid:48) . In the discrete setting, the evolution of the concentration c t ( m ) of particlesof mass m ∈ N ∗ at time t ≥ t c t ( m ) = 12 m − (cid:88) m (cid:48) =1 κ ( m, m (cid:48) ) c t ( m (cid:48) ) c t ( m − m (cid:48) ) − (cid:88) m (cid:48) ≥ κ ( m, m (cid:48) ) c t ( m ) c t ( m (cid:48) ) . (1.1)Norris considered in [24] far more general models of cluster coagulation , where therate of coalescence does not depend only on the mass of the particles but also onother parameters. In this general setting, most results on existence and uniquenessare obtained before a critical time, known as the gelation time , while the globalbehavior of the solutions after gelation, and in particular uniqueness, is not known.An example of a solvable cluster coagulation model is Bertoin’s model with limitedaggregations [3], which we shall simply call the model with arms . In this case,particles have a mass but also carry a certain number of potential links, called arms . Two particles of mass m and m (cid:48) may coagulate only if they have a positivenumber of arms, say a and a (cid:48) . When they coagulate, an arm of each is used tocreate the bond and both arms are then deactivated, hence creating a particle with Mathematics Subject Classification.
Primary: 34A34; Secondary: 82D60.
Key words and phrases.
Coagulation equations; Gelation; Generating functions; Method ofcharacteristics; Long-time behavior. a r X i v : . [ m a t h . C A ] S e p RAOUL NORMAND AND LORENZO ZAMBOTTI a + a (cid:48) − m + m (cid:48) . The coagulation rate of these two particles is aa (cid:48) .Therefore, if c t ( a, m ) is the concentration of particles with a ∈ N = { , , . . . } armsand mass m ∈ N ∗ = { , , . . . } , then the coagulation equation readsdd t c t ( a, m ) = 12 a +1 (cid:88) a (cid:48) =1 m − (cid:88) m (cid:48) =1 a (cid:48) ( a + 2 − a (cid:48) ) c t ( a (cid:48) , m (cid:48) ) c t ( a + 2 − a (cid:48) , m − m (cid:48) ) − (cid:88) a (cid:48) ≥ (cid:88) m (cid:48) ≥ aa (cid:48) c t ( a, m ) c t ( a (cid:48) , m (cid:48) ) . (1.2)For monodisperse initial concentrations, i.e. c ( a, m ) = { m =1 } µ ( a ), with µ =( µ ( a )) a ∈ N a measure on N with unit mean, it is proved in [3] that this equationhas a unique solution on some interval [0 , T ), where T = + ∞ if and only if K ≤ K := (cid:88) a ≥ a ( a − µ ( a ) . (1.3)In other words, if particles at time 0 have, on average, few arms, equation (1.2) hasa unique solution defined for all t ≥
0. When this is the case, as time passes, allavailable arms are used to create bonds and only particles with no arms remain inthe system. The limit concentrations c ∞ (0 , m ) as t → + ∞ of such particles turn outto be related to the distribution of the total population generated by a sub-criticalGalton-Watson branching process (see e.g. [2]) started from two ancestors: see [3, 4]and section 1.4 below.1.2. The gelation phase transition.
A formal computation shows that solutionsof (1.1) with multiplicative kernel κ ( m, m (cid:48) ) = mm (cid:48) should have constant mass M t := (cid:88) m ≥ mc t ( m ) , t ≥ , (1.4)i.e. ddt M t = 0. It is however well-known that if large particles can coagulate suf-ficiently fast, then one may observe in finite time a phenomenon called gelation ,namely the formation of particles of infinite mass, the gel . These particles do notcount in the computation of the mass so from the gelation time on, M t starts todecrease.The reason why (1.2) can be solved, is that it can be transformed into a solvablePDE involving the generating function of ( c t ) t ≥ . In Equation (1.1), this transfor-mation is also possible for several particular choices of the kernel κ ( m, m (cid:48) ), namelywhen κ is constant, additive or multiplicative: see e.g. [5]. In the multiplicative case κ ( m, m (cid:48) ) = mm (cid:48) , which is our main concern here, the total mass is a parameter of(1.1) and of the associated PDE, which is therefore easy to solve only when ( M t ) t ≥ is known. Existence and uniqueness of solutions of (1.1) are thus easy up to gelation,since in this regime, the total mass M t is constant.After gelation, the gel may or may not interact with the other particles. If itdoes, Equation (1.1) has to be modified into Flory’s equation (3.1). Else, the gel isinert, in which case Smoluchowski’s equation continues to hold. Obviously, they areidentical before gelation. OST-GELATION UNIQUENESS FOR COAGULATION EQUATIONS 3
Occurrence of gelation depends heavily on the choice of the coagulation rate κ ( m, m (cid:48) ), and in the multiplicative case, gelation always occurs [10, 12, 17]. Aftergelation, the mass is not known, so M t itself becomes an unknown of the equation,and well-posedness of the equation is then much less trivial. The multiplicative ker-nel is therefore particularly interesting, since it exhibits a non-trivial behavior butcan still be studied in detail by means of explicit computations.The same phenomenon of gelation has been observed in [3] for (1.2) for monodis-perse initial concentrations c . A formal computation shows that the the meannumber of arms A t A t := (cid:88) a,m ≥ a c t ( a, m ) , t ≥ , satisfies the equation ddt A t = − A t and should therefore be equal to t for all t ≥ / ( K −
1) if
K > ∞ if K ≤
1, where K is defined in (1.3).Again, the associated PDE is easy to solve before gelation since then, A t is known,while afterwards, the PDE contains the unknown parameter A t .1.3. Main result.
In this paper we investigate the global behavior of Smoluchow-ski’s equation with arms (1.2) before, at and after the gelation phase transition,proving existence and uniqueness of global solutions for a large class of initial con-ditions. The technique used, as in [3], is to transform the equation into a PDE.Since the total number of arms ( A t ) t ≥ is not a priori known, this PDE is non local,unlike the one obtained in the regime before gelation. This is the main difficulty wehave to deal with. We use a modification of the classical method of characteristicsto show uniqueness of solutions to this PDE, and hence to (1.2). We can considerinitial conditions ( c ( a, m ) , a ∈ N , m ∈ N ∗ ) with an initial infinite number of arms,that is, such that A := (cid:88) a,m ≥ ac ( a, m )is infinite, and show that there is a unique solution “coming down from infinitysufficiently fast”, i.e. such that, for positive t , (cid:90) t A s d s < + ∞ . Note however that this is no technical condition, but a mere assumption to ensurethat the equation is well-defined.We also consider a modification of this model which corresponds to Flory’s equa-tion for the model with arms. In this setting, the infinite mass particles, that is,the gel, interact with the other particles. We also prove existence, uniqueness andstudy the behavior of the solutions for this model.In both cases, our technique provides a representation formula allowing to com-pute various quantities, as the mean number of arms in the system and the limitingconcentrations. In Flory’s case, we extend to all possible initial concentrations thecomputations done in [3] in absence of gelation. In the first model, a slight modifi-cation appears which calls for a probabilistic interpretation; see section 1.4 below.
RAOUL NORMAND AND LORENZO ZAMBOTTI
This seems to be the first case of a cluster coagulation model for which globalwell-posedness in presence of gelation can be proven. Another setting to whichthese techniques could be applied is the coagulation model with mating introducedin [22].1.4.
Limiting concentrations.
In [3], explicit solutions to (1.2) are given formonodisperse initial conditions c ( a, m ) = µ ( a ) { m =1 } for some measure µ on N with unit first moment. In particular, when there is no gelation, i.e. K ≤ K is as in (1.3), and µ (cid:54) = δ , there are limiting concentrations c ∞ ( a, m ) = 1 m ( m − ν ∗ m ( m − { a =0 } , m ≥ , where ν ( m ) = ( m + 1) µ ( m + 1) is a probability measure on N different from δ .This formula clearly resembles the well-known formula of Dwass [7], which providesthe law of the total progeny T of a Galton-Watson process with reproduction law ν , started from two ancestors: P ( T = m ) = 2 m ν ∗ m ( m − , m ≥ . The similarity between the two formulas is no coincidence and is explained in [4] bymeans of the configuration model. For basics on Galton-Watson processes, see e.g.[2].Let us briefly explain the result of [4], referring e.g. to [26] for more resultson general random graphs. The configuration model aims at producing a randomgraph whose vertices have a prescribed degree. To this end, consider a number n of vertices, each being given independently a number of arms (that is, half-edges)distributed according to µ . Then, two arms in the system are chosen uniformly andindependently, and form an edge between the corresponding vertices. This procedureis repeated until there are no more available arms. Hence, one arrives to a final statewhich can be described as a collection of random graphs. Then Corollary 2 in [4] andthe discussion below show that, when there is no gelation, the proportion of trees ofsize m tends to c ∞ (0 , m ) when the number n of vertices tends to infinity. Hence, thefinal states in the configuration model and in Smoluchowski’s equation with armscoincide. This shows that the former is a good discrete model for coagulation.Interestingly, the absence-of-gelation condition K ≤ ν , i.e. toalmost sure extinction of the progeny, while K > c ∞ ( a, m ) = 1 m ( m − ν ∗ m ( m − { a =0 } , m ≥ , OST-GELATION UNIQUENESS FOR COAGULATION EQUATIONS 5 that is, the same explicit form as the one obtained in absence of gelation. Again, thisformula can be interpreted both in terms of a configuration model and of a super-critical Galton-Watson branching process. The relation between Flory’s equationwith arms and the configuration model is natural, since in both cases all particlesinteract with each other, no matter what their size is. It is worth noticing that, eventhough the limit concentrations have the same form with or without gelation, stillsome mass is eventually lost in presence of gelation, see (6.4) below.We also obtain the limiting concentrations for Smoluchowski’s equation with arms,namely c ∞ ( a, m ) = 1 m ( m − β m − ∞ ν ∗ m ( m − { a =0 } , where β ∞ is some constant, which is 1 when there is no gelation, and is greaterthan 1 otherwise, see Section 6.2. However, the probabilistic interpretation of β ∞ isunclear. One can recover Smoluchowski’s equation with arms from discrete modelsby preventing big particles from coagulating, as is done in [13] for the standardSmoluchowski equation, but the precise meaning of β ∞ still seems to require somelabor.1.5. Bibliographical comments.
Smoluchowski’s equation (1.1) has been exten-sively studied; we refer to the reviews [1, 18, 23]. Conditions on the kernel κ areknow for absence or presence of gelation, though this requires a precise definition ofgelation, see e.g. [11], or [14] in a probabilistic setting. For a general class of kernelsSmoluchowski’s solution has a unique solution before gelation [23, 6, 12, 18], and inthe multiplicative case gelation always occurs [10, 12, 17].For the monodisperse initial condition c ( m ) = { m =1 } , the first proof of existenceand uniqueness to (1.1) before gelation is given in [20], and a proof of global existenceand uniqueness can be found in [15]. The case of general nonzero initial conditionshas been considered by several papers in the Physics literature [8, 9, 19, 25, 31], andby at least one mathematical paper [27], which however treats in full details onlythe regime before gelation, see Remark 2.7 below. The same authors also providein [28] an exact formula for the post-gelation mass of (1.1), but with no rigorousproof.Thus, a clear statement about well-posedness of (1.1) for the most general initialconditions still seems to be missing, and our paper tries to fill this gap. We adaptthe classical method of characteristics for generating functions, see [5, 3], whichyields easily uniqueness before gelation for a multiplicative kernel [21]. We can inparticular consider initial concentrations with infinite total mass, i.e. such that M := (cid:90) (0 , + ∞ ) m c (d m ) = + ∞ , as long as (cid:82) (0 , + ∞ ) ( m ∧ c (d m ) < + ∞ . This covers for instance initial conditionsof the type c (d m ) = C p m − p d m with p ∈ [1 , RAOUL NORMAND AND LORENZO ZAMBOTTI
However, the case of an infinite initial mass seems to have been considered only in[16] in the discrete case, so we refer to Section 2.4 below for a proof.1.6.
Plan of the article.
We start off in Section 2 by considering existence, unique-ness and representation formulas for global solutions of (1.1), introducing and ex-ploiting all main techniques which are needed afterwards to tackle the same issuesin the case of (1.2). We prove that for the most general initial conditions µ (d m ), apositive measure on (0 , + ∞ ), Smoluchowski’s equation with a multiplicative kernelhas a unique solution before and after gelation. We also show existence and unique-ness for the modified version of Smoluchowski’s model, namely Flory’s equation, inSection 3. The techniques used are generalized in Section 4 and 5, where we proveanalogous results for the models with arms. We compute the limiting concentrationsin Section 6, which are not trivial, in comparison with the standard Smoluchowskiand Flory cases, for which they are always zero.2. Smoluchowski’s equation
In this section we develop our method in the case of equation (1.1), provingexistence, uniqueness and representation formulas for global solutions. Let us firstfix some notations. • M + f is the set of all non-negative finite measures on (0 , + ∞ ). • M + c is the set of all non-negative Radon measures on (0 , + ∞ ). • For µ ∈ M + c and f ∈ L ( µ ) or f ≥ (cid:104) µ, f (cid:105) = (cid:90) (0 , + ∞ ) f ( m ) µ (d m ) . We will write m for the function m (cid:55)→ m , m for m (cid:55)→ m , etc. • For φ : (0 , + ∞ ) → R and m, m (cid:48) >
0, ∆ φ ( m, m (cid:48) ) = φ ( m + m (cid:48) ) − φ ( m ) − φ ( m (cid:48) ). • C c (0 , + ∞ ) is the space of continuous functions on (0 , + ∞ ) with compactsupport. • For a function ( t, x ) (cid:55)→ φ t ( x ), φ (cid:48) t ( x ) is the partial derivative of φ with respectto x . • ∂ + ∂t or d + d t denotes the right partial derivative with respect to t .We are interested in Smoluchowski’s equation (1.1) with multiplicative coagulationkernel κ ( m, m (cid:48) ) = mm (cid:48) . Note that the second requirement in the following definitionis only present for the equation to make sense. Definition 2.1.
Let µ ∈ M + c . We say that a family ( µ t ) t ≥ ⊂ M + c solves Smolu-chowski’s equation if • for every t > , (cid:82) t (cid:104) µ s (d m ) , m (cid:105) d s < + ∞ , • for all φ ∈ C c (0 , + ∞ ) and t > (cid:104) µ t , φ (cid:105) = (cid:104) µ , φ (cid:105) + 12 (cid:90) t (cid:104) µ s (d m ) µ s (d m (cid:48) ) , mm (cid:48) ∆ φ ( m, m (cid:48) ) (cid:105) d s, (2.1) • if (cid:104) µ , m (cid:105) < + ∞ , then t (cid:55)→ (cid:104) µ t , m (cid:105) is bounded in a right neighborhood of0. OST-GELATION UNIQUENESS FOR COAGULATION EQUATIONS 7
The global behavior of this equation has been studied first for monodisperse initialconditions (i.e. µ = δ ), in which case it can be proven that there is a uniquesolution ( µ t ) t ≥ on R + , which is also explicit, see [20, 15]. This solution clearlyexhibits the gelation phase transition. Up to the gelation time T gel = 1, the totalmass (cid:104) µ t , m (cid:105) is constant and equal to 1, and then it decreases: (cid:104) µ t , m (cid:105) = 1 /t for t ≥
1. Moreover, the second moment (cid:104) µ t , m (cid:105) is finite before time 1, and theninfinite on [1 , + ∞ ). It is also known in the literature that for any nonzero initialconditions, there is a gelation time 0 < T gel < + ∞ , such that there is a uniquesolution to (2.1) on [0 , T gel ), and (cid:104) µ t , m (cid:105) → + ∞ when t → T − gel : see e.g. [12]. Theorem 2.2.
Let µ ∈ M + c a non-null measure such that (cid:104) µ , m ∧ (cid:105) = (cid:90) (0 , + ∞ ) ( m ∧ µ (d m ) < + ∞ . (2.2) We can then define M := (cid:104) µ , m (cid:105) ∈ (0 , + ∞ ] , K := (cid:104) µ , m (cid:105) ∈ (0 , + ∞ ] , and the function g ( x ) := (cid:104) µ , mx m (cid:105) = (cid:90) (0 , + ∞ ) mx m µ (d m ) , x ∈ [0 ,
1] (2.3) with g (1) = M ∈ (0 , + ∞ ] . Let T gel := 1 /K ∈ [0 , + ∞ ) . (2.4) Then Smoluchowski’s equation (2.1) has a unique solution on R + . It has the follow-ing properties.(1) The total mass M t = (cid:104) µ t , m (cid:105) is continuous on [0 , + ∞ ) . It is constant on [0 , T gel ] and strictly decreasing on [ T gel , + ∞ ) . It is analytic on R + \{ T gel } .(2) If the following limit exists ν := − lim x → − ( g (cid:48) ( x )) g (cid:48) ( x ) + xg (cid:48)(cid:48) ( x ) ∈ [ −∞ , , then the right derivative ˙ M T gel of M at t = T gel is equal to ν .(3) Let m = inf supp µ ∈ [0 , + ∞ ) . When t → + ∞ , tM t → m . (4) The second moment (cid:104) µ t , m (cid:105) is finite for t ∈ [0 , T gel ) and infinite for t ∈ [ T gel , + ∞ ) . Remark 2.3. • This result allows to recover the pre- and post-gelation for-mulas obtained with no rigorous proof in some earlier papers [9, 8, 15, 19,27, 28, 25]. The decrease of the mass in 1 /t when m > /t for the mass were provenin [11, 17]. RAOUL NORMAND AND LORENZO ZAMBOTTI • If m = 0, the mass tends to 0 more slowly than 1 /t : small particles need tocoagulate before any big particle can appear, and they coagulate really slowly.For instance, a straightforward computation shows that if µ (d m ) = e − m d m ,then M t ∼ t − / . More generally, the explicit formula in Proposition (2.6)allows to compute M t for any initial conditions. • With this formula, it is easy to check that ˙ M T gel + can be anything from −∞ to 0. For instance, ˙ M T gel + = 0 for g ( x ) = (1 − x ) log(1 − x ) + x , ˙ M = −∞ for g ( x ) = √ − x log(1 − x ) + x , and for 0 < α < + ∞ , ˙ M = − α for g ( x ) = 1 − √ − x α . In particular, M need not be convex on [ T gel , + ∞ ).2.1. Preliminaries.
Let µ be defined as in the previous statement. We shallprove that, starting from µ , there is a unique solution to (2.1) on R + , and givea representation formula for this solution. This allows to study the behavior of themoments. Let us start with some easy lemmas. So take a solution ( µ t ) t ≥ to (2.1)and set M t = (cid:104) µ t , m (cid:105) . (2.5)The two following lemmas are easy to prove, using monotone and dominated con-vergence. Lemma 2.4. ( M t ) t ≥ is monotone non-increasing and right-continuous. Moreover, M t < + ∞ for all t > .Proof. Take φ K ( m ) = m for m ∈ [0 , K ], φ K ( m ) = 2 K − m for m ∈ [ K, K ], and φ K ( m ) = 0 for m ≥ K , so that φ K ∈ C c . Plugging φ K in Smoluchowski’s equation(2.1), letting K → + ∞ and using Fatou’s lemma readily shows that ( M t ) t ≥ ismonotone non-increasing. Note also that t (cid:55)→ M t = sup K (cid:104) µ t , φ K (cid:105) is the supremumof a sequence of continuous functions and so is lower semi-continuous, which implies,for a monotone non-increasing function, right-continuity. Finiteness of M t is nowobvious since s (cid:55)→ M s , and hence s (cid:55)→ M s , are integrable by Definition 2.1. (cid:3) Lemma 2.5.
Assume that t (cid:55)→ (cid:104) µ t , m (cid:105) is bounded on some interval [0 , T ] . Then M t = M for t ∈ [0 , T ] . By Lemma 2.4, (cid:104) µ t , m (cid:105) < + ∞ for t >
0, so that we can define g t ( x ) = (cid:104) µ t , mx m (cid:105) = (cid:90) (0 , + ∞ ) mx m µ t (d m ) , x ∈ [0 , , t > , (2.6)which is the generating function of mµ t (d m ). Then, using a standard approximationprocedure, it is easy to see that g satisfies g t ( x ) = g ( x ) + (cid:90) t x ( g s ( x ) − M s ) ∂ + g s ∂x ( x ) d s, t ≥ , x ∈ (0 , g t (1) = M t , t ≥ . (2.7)It is well-known, and will be proven again below, that M t = M for all t ≤ T gel , sincethen, the PDE (2.7) can be solved by the method of characteristics: the function φ t ( x ) : [0 , (cid:55)→ [0 , φ t ( x ) = xe t ( M − g ( x )) , x ∈ [0 , , t ≤ T gel OST-GELATION UNIQUENESS FOR COAGULATION EQUATIONS 9 is one-to-one and onto, has an inverse h t : [0 , (cid:55)→ [0 ,
1] and we find g t ( x ) = g ( h t ( x )) , x ∈ [0 , , t ≤ T gel . However M t is not necessary constant for t > T gel and the form of φ t has to bemodified; we thus define φ t ( x ) = xα t e − tg ( x ) , x ∈ [0 , , t > α t := exp (cid:18)(cid:90) t M s d s (cid:19) , t ≥ . (2.9)For t > T gel , M t is possibly less than M and φ t , which depends explicitly on( M s ) s ∈ [0 ,t ] , is possibly neither injective nor surjective. We shall prove that it isindeed possible to find (cid:96) t ∈ (0 ,
1) such that φ t ( x ) : [0 , (cid:55)→ [0 , (cid:96) t ] is one-to-one and (cid:96) t is uniquely determined by g .2.2. Uniqueness of solutions.
Using an adaptation of the method of characteris-tics, we are going to prove the following result. Note that in [27], this properties areclaimed to be true but a proof seems to lack. We will use the same techniques in theproof of Theorem 4.2 for the model with arms, but they are easier to understand inthe present case.
Proposition 2.6.
Let ( µ t ) t ≥ be a solution of Smoluchowski’s equation (2.1) .(1) For all t ∈ [0 , T gel ] , M t = M = g ( (cid:96) t ) , where (cid:96) t := 1 . For all t > T gel , M t = g ( (cid:96) t ) where (cid:96) t ∈ (0 , is uniquely defined by (cid:96) t g (cid:48) ( (cid:96) t ) = 1 t . (2.10) Moreover (cid:96) t and φ t ( · ) satisfy φ (cid:48) t ( (cid:96) t ) = 0 , φ t ( (cid:96) t ) = 1 > φ t ( x ) , ∀ x ∈ (0 , . (2.11) (2) For all t > , the function φ t ( · ) defined in (2.8) has a right inverse h t : [0 , (cid:55)→ [0 , (cid:96) t ] , φ t ( h t ( x )) = x, x ∈ [0 , , (2.12) and g t ( x ) = g ( h t ( x )) , t > , x ∈ [0 , . (2.13) (3) The functions ( (cid:96) t ) t ≥ and ( M t ) t ≥ are continuous.(4) ( µ t ) t ≥ is uniquely defined by µ . Remark 2.7. • For all t ≤ T gel , M t = M , (cid:96) t = 1 and φ t : [0 , (cid:55)→ [0 , t > T gel , (cid:96) t <
1, i.e. there is indeed x ∈ [0 ,
1] such that φ t ( x ) = 1, see Lemma2.9; the second one, is that (cid:96) t = m t , i.e. φ t ( · ) has an absolute maximumat (cid:96) t , see Lemma 2.10. In other words, one has to exclude the dotted linesas possible profiles of φ t ( · ) in Figure 1. These properties are not obvious,since φ t depends on ( M s ) s ∈ [0 ,t ] which is, at this point, unknown. All otherproperties are derived from these two. Figure 1. φ t before and after gelation. The dotted lines representwhat φ t may look like. The solid one is the actual φ t . • In [27, Section 6] one finds a discussion of post-gelation solutions, in par-ticular of the results of our Proposition 2.6. However this discussion fallsshort of a complete proof, since the two above-mentioned properties are notproven. In particular, no precise statement about what initial conditions canbe considered is given.The following lemma is a list of obvious but useful properties satisfied by g and φ . Lemma 2.8.
The function g defined in (2.6) satisfies the following properties. (a1) ( t, x ) (cid:55)→ g t ( x ) is finite and continuous on [0 , + ∞ ) × [0 , ; (a2) For all x ∈ [0 , , t (cid:55)→ g t ( x ) is right differentiable on (0 , + ∞ ) ; (a3) For all t ≥ , x (cid:55)→ g t ( x ) is analytic on (0 , and monotone non-decreasing; (a4) For all t > , x (cid:55)→ g t ( x ) ∈ [0 , + ∞ ] is continuous on [0 , .The function φ defined in (2.8) satisfies the following properties. (b1) φ t is continuous on [0 , and analytic on (0 , ; (b2) φ t (0) = 0 , φ t (1) = e − (cid:82) t ( M − M s ) d s ∈ [0 , ; (b3) φ (cid:48) t ( x ) = α t e − tg ( x ) (1 − txg (cid:48) ( x )) for x ∈ (0 , ; (b4) For t ≤ T gel , φ t is increasing. For t > T gel , x (cid:55)→ xg (cid:48) ( x ) is increasing, φ (cid:48) t (0) > and φ (cid:48) t (1) < . In particular, for t > T gel , there is precisely onepoint m t ∈ (0 , such that φ (cid:48) t ( m t ) = 0 . (2.14)(b5) For t > T gel , φ t is increasing on [0 , m t ] and decreasing on [ m t , .Moreover, (c1) The map ( t, x ) (cid:55)→ φ t ( x ) is continuous on R + × [0 , ; (c2) The map ( t, x ) (cid:55)→ φ (cid:48) t ( x ) is continuous on R + × (0 , ; (c3) For every x ∈ [0 , , t (cid:55)→ φ t ( x ) is right differentiable and ∂ + φ t ∂t = φ t ( x )( M t − g ( x )) x ∈ [0 , , t ≥ . (2.15) OST-GELATION UNIQUENESS FOR COAGULATION EQUATIONS 11
Property (b5) implies that there are at most two points in (0 ,
1) where φ t equals1. Take (cid:96) t to be the smallest, if any, i.e. (cid:96) t = inf { x ≥ φ t ( x ) = 1 } (inf ∅ := 1) . (2.16) Lemma 2.9. (1) For every t ≥ and every x ∈ [0 , (cid:96) t ] g t ( φ t ( x )) = g ( x ) . (2.17) (2) For all t ∈ [0 , T gel ] , (cid:96) t = 1 , and for t > T gel , < (cid:96) t < . In particular, for all t > , φ t ( (cid:96) t ) = 1 and g ( (cid:96) t ) = g t (1) = M t . (2.18) (3) Finally, t (cid:55)→ (cid:96) t is monotone non-increasing and continuous on R + .Proof. (1) Let us first prove that there exists τ > t ∈ [0 , τ [. Fix 0 < a < b <
1. Since 0 < min [ a,b ] φ < max [ a,b ] φ <
1, then byproperty (c1) there is τ > < min [ a,b ] φ t < max [ a,b ] φ t < , ∀ t ∈ [0 , τ ) . So, for a fixed x ∈ [ a, b ], the function u t := g t ( φ t ( x )) − g ( x )is well-defined and using (2.7) and (2.15), we see that u t = (cid:90) t (cid:18) ∂ + g s ∂s ( φ s ( x )) + ∂g s ∂x ( φ s ( x )) ∂ + φ s ∂s ( x ) (cid:19) d s = (cid:90) t γ s u s d s where γ t := ∂g t ∂x ( φ t ( x )) φ t ( x ) , t > . Since x ∈ [0 , t ∈ [0 ,τ ) | γ t | < + ∞ and therefore u t ≡
0. Hence (2.17)holds for x ∈ [ a, b ] and t ∈ [0 , τ [. Since both terms of (2.17) are analyticfunctions of x on (0 , (cid:96) t ), by analytic continuation, (2.17) actually holds on(0 , (cid:96) t ), and hence on [0 , (cid:96) t ] by continuity.(2) Let us now extend this formula to t ∈ R + . Let T = sup { t > ∀ s ∈ [0 , t ] , ∀ x ∈ [0 , (cid:96) s ] , g s ( φ s ( x )) = g ( x ) } ≥ τ > , assume T < + ∞ , and denote by (cid:96) the left limit of ( (cid:96) t ) t ≥ at T . First, (cid:96) cannot be 0, since otherwise we would get when s → T − φ s ( (cid:96) s ) = (cid:96) s α s e − sg ( (cid:96) s ) → . For every t < T − , 0 < (cid:96) ≤ (cid:96) t , so for every x ∈ (0 , (cid:96) ), g t ( φ t ( x )) = g ( x ) and φ t ( x ) <
1. Using the continuity property (c1) and passing to the limit when t → T − in this equality, we get g T ( φ T ( x )) = g ( x ) , ∀ x ∈ (0 , (cid:96) ) . By the same reasoning as in point (i), we obtain a T (cid:48) > T such that g t ( φ t ( x )) = g ( x ) for all t ∈ [ T, T (cid:48) ) and x in a non-empty open subset of(0 , (cid:96) ). By analyticity and continuity, the formula g t ( φ t ( x )) = g ( x ) holds for every t ∈ [ T, T (cid:48) ) and x ∈ [0 , (cid:96) t ]. This contradicts the definition of T , and so T = + ∞ . This concludes the proof of point (1) of the Lemma.(3) For the statement (2) of the Lemma, let us show first that (cid:104) µ t , m (cid:105) is boundedon [0 , T ), for every T ∈ [0 , T gel ). Let T (cid:48) , the smallest time when this fails(provided of course that T gel > T (cid:48) >
0. Differentiating (2.17) with respect to x and having x tend to (cid:96) t = 1gives, for t < T (cid:48) , g (cid:48) t (1) = (cid:104) µ t , m (cid:105) = 11 − tK . This quantity explodes only when t = T gel = 1 /K , so T (cid:48) = T gel .(4) The boundedness of ( (cid:104) µ t , m (cid:105) ) t ∈ [0 ,T ) just proven for all T ∈ [0 , T gel ) andLemma 2.5 imply that for t ∈ [0 , T gel ), M t = M . By the definition (2.8) of φ t , it follows that φ t (1) = 1 for t ∈ [0 , T gel ). But φ t is increasing, so (cid:96) t = 1for t ∈ [0 , T gel ). Assume now that for some t > T gel , (cid:96) t = 1. Then (2.17)holds on [0 , x , whereas the left one decreases in a left neighborhood of 1 since φ (cid:48) t (1) <
0. The fact that φ t ( (cid:96) t ) = 1 follows then directly from the definitionof (cid:96) t and the continuity of φ t ( · ). Finally, the inequality (cid:96) t > φ t (0) = 0, and computing (2.17) at x = (cid:96) t gives (2.18). This concludesthe proof of (2).(5) We know that (cid:96) t = 1 and M t = M for all t < T gel . Now, g is strictlyincreasing and continuous. Since ( M t ) t ≥ is monotone non-increasing andright-continuous by Lemma 2.4, so is ( (cid:96) t ) t ≥ by (2.18). To get left-continuityof ( (cid:96) t ) t>T gel , consider t > T gel , and let (cid:96) be the left limit of (cid:96) s at t . We have (cid:96) ≤ (cid:96) t +( t − T gel ) / <
1, so by the continuity property (c1) above,1 = φ s ( (cid:96) s ) → s → t − φ t ( (cid:96) ) . Hence φ t ( (cid:96) ) = 1. Assume (cid:96) > (cid:96) t (that is, (cid:96) is the second point where φ t reaches 1). Take x ∈ ( (cid:96) t , (cid:96) ). By property (b5), φ t ( x ) >
1. But on the otherhand, x < (cid:96) ≤ (cid:96) s for s < t , so φ s ( x ) ≤
1, and so φ t ( x ) ≤
1, and this is acontradiction. So (cid:96) = (cid:96) t and ( (cid:96) t ) t ≥ is indeed continuous. This concludes theproof of (3) and of the Lemma. (cid:3) Finally, we will see that for t > T gel , (cid:96) t = m t , so that φ t increases from 0 to1, which is its maximum, and then decreases. To this end, recall that ( (cid:96) t ) t ≥ ismonotone non-increasing and that ( (cid:96) t ) t ≥ and ( φ t ) t ≥ are continuous, so the chainrule for Stieltjes integrals and (2.15) give1 = φ t ( (cid:96) t ) = φ ( (cid:96) ) + (cid:90) t φ (cid:48) s ( (cid:96) s ) d (cid:96) s + (cid:90) t ∂ + φ s ∂s ( (cid:96) s ) d s = 1 + (cid:90) t φ (cid:48) s ( (cid:96) s ) d (cid:96) s + (cid:90) t φ s ( (cid:96) s )( M s − g ( (cid:96) s )) d s OST-GELATION UNIQUENESS FOR COAGULATION EQUATIONS 13 that is, with (2.18), φ (cid:48) t ( (cid:96) t ) d (cid:96) t = 0 . (2.19)Hence, d (cid:96) t -a.e φ (cid:48) t ( (cid:96) t ) = 0, i.e. (cid:96) t = m t . This is actually true for all t > T gel , aswe shall now prove. This result also has its counterpart in the model with arms,namely part 3 of the proof of Theorem 4.2. Lemma 2.10.
For every t > T gel , φ (cid:48) t ( (cid:96) t ) = 0 , i.e. (cid:96) t = m t , the point where φ t attains its maximum. In particular, (cid:96) t g (cid:48) ( (cid:96) t ) = 1 t , ∀ t > T gel . (2.20) Proof.
First, recall that φ t is increasing on [0 , (cid:96) t ], so that φ (cid:48) t ( (cid:96) t ) ≥
0, that is (cid:96) t g (cid:48) ( (cid:96) t ) ≤ t . (2.21)Assume now that there is a t > T gel such that φ (cid:48) t ( (cid:96) t ) >
0, and consider s = sup { r ∈ ( T gel , t ) : φ (cid:48) r ( (cid:96) r ) = 0 } . As noted before, t (cid:55)→ (cid:96) t is strictly decreasing for t > T gel for any t > T gel , sod (cid:96) t ([ T gel , T gel + ε [) > ε >
0. Hence there are points r < t where φ (cid:48) r ( (cid:96) r ) = 0,and thus the definition of s does make sense.Take now ( r n ) a sequence of points such that T < r n < t , φ (cid:48) r ( (cid:96) r ) = 0 and ( r n )converges to s . Since 0 < (cid:96) s <
1, by property (c2) above, we get0 = φ (cid:48) r n ( (cid:96) r n ) → φ (cid:48) s ( (cid:96) s )so that φ (cid:48) s ( (cid:96) s ) = 0. This shows that s < t , and that for r ∈ ( s, t ), φ (cid:48) r ( (cid:96) r ) >
0. Hence,by continuity of ( (cid:96) r ) r ≥ and by (2.19), ( (cid:96) r ) r ∈ [ s,t ] is constant. This gives1 s = (cid:96) s g (cid:48) ( (cid:96) s ) = (cid:96) t g (cid:48) ( (cid:96) t ) ≤ t which is a contradiction since s < t . In particular, φ (cid:48) t ( (cid:96) t ) = 0 implies (2.20). (cid:3) Proof of Proposition 2.6.
By Lemma 2.10, necessarily M t = M on [0 , T gel ] and for t > T gel , M t := g t (1) = g ( (cid:96) t ), where (cid:96) t g (cid:48) ( (cid:96) t ) = 1 t . (2.22)Since x (cid:55)→ xg (cid:48) ( x ) is strictly increasing from [0 ,
1] to [0 , K ], where K = (cid:104) µ , m (cid:105) =1 /T gel , this equation has a unique solution for t > T gel . Hence M t is uniquely defined.Therefore α t and φ t are uniquely determined by g , so we can define φ t as in (2.8),and Lemma 2.9 shows that g t ( φ t ( x )) = g ( x ) for x ∈ [0 , (cid:96) t ], and that φ t is a bijectionfrom [0 , (cid:96) t ] to [0 , h t , and compounding by h t in theprevious formula gives g t ( x ) = g ( h t ( x )) (2.23)for all x ∈ [0 , t ≥
0. Thus g t can be expressed by a formula involving only g andin particular, ( µ t ) t ≥ depends only on µ . This shows the uniqueness of a solutionto Smoluchowski’s equation (2.1). (cid:3) Behavior of the moments.
In this paragraph, we will study the behaviorof the first and second moment of ( µ t ) t ≥ as time passes, showing how to proverigorously and recover the results of [9]. For more general coagulation rates, one canobtain upper bounds of the same nature, see [17].First consider the mass M t = (cid:104) µ t , m (cid:105) . We will always assume that T gel < + ∞ .Let us start with the following lemma. Lemma 2.11.
Let ν ∈ M + c be a measure which integrates x (cid:55)→ y x for small enough y > . Let m be the infimum of its support. Then lim y → + (cid:104) ν, xy x (cid:105)(cid:104) ν, y x (cid:105) = m . Proof.
First, note that xy x ≥ my x ν -a.e. solim inf y → (cid:104) ν, xy x (cid:105)(cid:104) ν, y x (cid:105) ≥ m . Let us prove now that lim sup y → (cid:104) ν, xy x (cid:105)(cid:104) ν, y x (cid:105) ≤ m . Assume this is not true. Then, up to extraction of a subsequence, we may assumethat there exists α > y ∈ (0 , (cid:104) ν, xy x (cid:105) ≥ ( m + α ) (cid:104) ν, y x (cid:105) . Hence (cid:104) ν, ( x − m − α ) y x (cid:105) ≥
0, so (cid:104) ν, ( x − m − α ) y x { x>m + α } (cid:105) ≥ (cid:104) ν, ( m + α − x ) y x { m ≤ x ≤ m + α } (cid:105) . (2.24)But (cid:104) ν, ( m + α − x ) y x { m ≤ x ≤ m + α } (cid:105) ≥ (cid:104) ν, ( m + α − x ) y x { m ≤ x ≤ m + α/ } (cid:105)≥ (cid:104) ν, ( m + α − x ) { m ≤ x ≤ m + α/ } (cid:105) y m + α/ and (cid:104) ν, ( x − m − α ) y x { x>m + α } (cid:105) ≤ (cid:104) ν, ( x − m − α ) { x>m + α } (cid:105) y m + α . With (2.24), this shows that (cid:104) ν, ( x − m − α ) { x>m + α } (cid:105) y α/ ≥ (cid:104) ν, ( m + α − x ) { m ≤ x ≤ m + α/ } (cid:105) and having y tend to zero gives0 ≥ (cid:104) ν, ( m + α − x ) { m ≤ x ≤ m + α/ } (cid:105) which is a contradiction since ν ([ m , m + α/ > (cid:3) Corollary 2.12.
The mass of the system is continuous and positive. It is strictlydecreasing on [ T gel , + ∞ ) . Moreover, denote m = inf supp µ . Then lim t → + ∞ tM t = m . OST-GELATION UNIQUENESS FOR COAGULATION EQUATIONS 15
Proof.
Recall that M t = g ( (cid:96) t ) so the first properties follow from Lemma 2.9. Denotenow ν (d m ) = mµ (d m ). For t > T gel , (cid:96) t g (cid:48) ( (cid:96) t ) = 1 /t , so1 tM t = (cid:104) ν, x(cid:96) xt (cid:105)(cid:104) ν, (cid:96) xt (cid:105) and since (cid:96) t → t → + ∞ , this tends to m by Lemma 2.11. (cid:3) We can also study the behavior of the mass for small times. Recall that beforegelation, the mass is constant at 1. We have seen that it is continuous at the gelationtime. We may then wonder if its derivative is continuous, that is if ˙ M T gel + is zero ornot. Lemma 2.13.
The right derivative of M at T gel is given by ˙ M T gel + = − lim x → − g (cid:48) ( x ) g (cid:48) ( x ) + xg (cid:48)(cid:48) ( x ) ∈ [ −∞ , provided the limit exists.Proof of Lemma 2.13. For t > T gel , f ( (cid:96) t ) = 1 /t with f ( x ) = xg (cid:48) ( x ), and 0 < (cid:96) t < f (cid:48) ( (cid:96) t ) (cid:54) = 0, so by the inverse mapping theorem, ( (cid:96) t ) t ≥ is differentiable and˙ (cid:96) t = − t f (cid:48) ( (cid:96) t ) . Using the fact that M t = g ( (cid:96) t ), it is then easy to see that˙ M t = − (cid:96) t g (cid:48) ( (cid:96) t ) g (cid:48) ( (cid:96) t ) + (cid:96) t g (cid:48)(cid:48) ( (cid:96) t ) . Since ( (cid:96) t ) t ≥ is continuous at T gel and (cid:96) T gel = 1, the result follows. (cid:3) Recall that the gelation time is precisely the first time when the second moment (cid:104) µ t , m (cid:105) of ( µ t ) t ≥ becomes infinite. It actually remains infinite afterwards. Corollary 2.14.
For all t ≥ T gel , (cid:104) µ t , m (cid:105) = + ∞ .Proof. Note that (cid:104) µ t , m (cid:105) = g (cid:48) t (1) , this formula being understood as a monotone limit. By (2.17), for x < (cid:96) t φ (cid:48) t ( x ) g (cid:48) t ( φ t ( x )) = g (cid:48) ( x ) . When x → (cid:96) − t , φ (cid:48) t ( x ) → g (cid:48) ( x ) → g (cid:48) ( (cid:96) t ) (cid:54) = 0 since (cid:96) t >
0. So g (cid:48) t ( φ t ( (cid:96) t )) = g (cid:48) t (1) = + ∞ . (cid:3) Existence of solutions.
Existence of solutions of (2.1) is a well-known topic,see e.g. [13]. However, the case M = + ∞ is apparently new, so that we give ashort proof for the general case based on previous papers, mainly [27].Let now µ ∈ M + f be as in the statement of Theorem 2.2 and let us set g as in(2.3), (cid:96) t and M t as in point (1) of Proposition 2.6, α t and φ t as in (2.9) and (2.8).Then it is easy to see that φ t admits a right inverse h t satisfying (2.12), and we canthus define g t ( x ) := g ( h t ( x )) , t ≥ , x ∈ [0 , . It is an easy but tedious task to check that g t satisfies (2.7) and all properties (a1)-(a4) above. In particular, if g (1) = + ∞ then h t (1) < g t (1) < + ∞ for all t >
0. Following [27], we can now prove the following.
Proposition 2.15.
For all t > there exists µ t ∈ M + f such that g t ( x ) = (cid:104) µ t , mx m (cid:105) = (cid:90) (0 , + ∞ ) mx m µ t (d m ) , x ∈ [0 , . Proof.
Let t > y ≥ y ) := g ( e − y ) , Γ( y ) := tg ( e − y ) , G ( y ) := Γ( y ) + y − log α t = − log φ t ( e − y ) . We recall that f : [0 , + ∞ ) (cid:55)→ [0 , + ∞ ) is completely monotone if f is continuous on[0 , + ∞ ), infinitely many times differentiable on (0 , + ∞ ) and( − k d k fdy k ( y ) ≥ , ∀ k ≥ , y ∈ (0 , + ∞ ) . It is easy to see that Φ and Γ are completely monotone. Moreover, G has a rightinverse G − : [0 , + ∞ ) (cid:55)→ [log(1 /(cid:96) t ) , + ∞ ) , G − ( y ) = − log h t ( e − y ) , y ≥ , and therefore by the definitions g ( h t ( e − y )) = Φ( G − ( y )) , y ≥ . By [27, Thm. 3.2], Φ ◦ G − is completely monotone and therefore, by Bernstein’sTheorem, there exists a unique ν t ∈ M + f such that g t ( e − y ) = g ( h t ( e − y )) = Φ( G − ( y )) = (cid:90) (0 , + ∞ ) e − ym ν t (d m ) , y ≥ . Since g t (1) < + ∞ for all t >
0, we obtain that (cid:104) ν t , m (cid:105) < + ∞ , so that we can set µ t (d m ) := m ν t (d m ), and we have found that there is a unique µ t ∈ M + f such that g t ( x ) = g ( h t ( x )) = (cid:90) (0 , + ∞ ) x m m µ t (d m ) , x ∈ (0 , . (cid:3) In order to show that ( µ t ) t ≥ is a solution of Smoluchowski’s equation in the senseof Definition 2.1, we have to check that (cid:82) ε M t d t < + ∞ for all ε >
0. This is thecontent of the next result.
OST-GELATION UNIQUENESS FOR COAGULATION EQUATIONS 17
Lemma 2.16. If ( µ t ) t ≥ is the family constructed in Proposition 2.15, then for all ε > , (cid:82) ε (cid:104) µ s , m (cid:105) d s < + ∞ .Proof. If M < + ∞ then there is nothing to prove, since ( M t ) t ≥ is monotone non-increasing, so let us consider the case M = + ∞ and thus T gel = 0. Since M t = g ( (cid:96) t )is bounded and continuous for t ∈ [ δ, ε ] for all δ ∈ (0 , ε ), we have by (2.10) and (2.3) (cid:90) εδ M t d t = (cid:90) εδ g ( (cid:96) t ) d t = εg ( (cid:96) ε ) − δg ( (cid:96) δ ) − (cid:90) εδ tg ( (cid:96) t ) g (cid:48) ( (cid:96) t ) d (cid:96) t ≤ εg ( (cid:96) ε ) − (cid:90) εδ g ( (cid:96) t ) d (cid:96) t (cid:96) t = εg ( (cid:96) ε ) + 2 (cid:90) (cid:96) δ (cid:96) ε g ( y ) d yy ≤ εg ( (cid:96) ε ) + 2 (cid:96) ε (cid:104) µ , m m (cid:105) ≤ εg ( (cid:96) ε ) + 2 (cid:96) ε (cid:104) µ , m ∧ (cid:105) . Letting δ ↓
0, by (2.2) we obtain the desired result. (cid:3)
We now finish the proof of existence of a solution by showing that ( µ t ) t ≥ indeedsolves (2.1). By choosing x = e − y , y ≥
0, in (2.7), we find an equality betweenLaplace transforms. Since the Laplace transform is one-to-one, then we obtain(2.1).
Remark 2.17.
In the proof of uniqueness, we may only require that (cid:104) µ , my m (cid:105) < + ∞ for every y ∈ [0 , (cid:104) µ , m ∧ (cid:105) = + ∞ , then (cid:82) t M s d s = + ∞ for all t >
0, in contradiction with Definition 2.1 of a solution.3.
Flory’s equation
We will now consider the modified version of Smoluchowski’s equation, also knownas
Flory’s equation , with a multiplicative kernel.
Definition 3.1.
Let µ ∈ M + c . We say that a family ( µ t ) t ≥ ⊂ M + c solves Flory’sequation (2.1) if • for every t > , (cid:82) t (cid:104) µ s (d m ) , m (cid:105) d s < + ∞ , • for all φ ∈ C c (0 , + ∞ ) and t > (cid:104) µ t , φ (cid:105) = (cid:104) µ , φ (cid:105) + 12 (cid:90) t (cid:104) µ s (d m ) µ s (d m (cid:48) ) , mm (cid:48) ∆ φ ( m, m (cid:48) ) (cid:105) d s − (cid:90) t (cid:104) µ s , φ (cid:105)(cid:104) µ (d m ) − µ s (d m ) , m (cid:105) d s, (3.1) • if (cid:104) µ , m (cid:105) < + ∞ , then t (cid:55)→ (cid:104) µ t , m (cid:105) is bounded in a right neighborhood of0. In equation (3.1), the mass that vanishes in the gel interacts with the other par-ticles. It is a modified Smoluchowski’s equation, where a term has been added,representing the interaction of the particles of mass m with the gel, whose mass is (cid:104) µ − µ s , m (cid:105) i.e. precisely the missing mass of the system. Notice that in this case the equationmakes sense only if (cid:104) µ , m (cid:105) < + ∞ .The mass is expected to decrease faster in this case than for (2.1). This is actuallytrue, as we can see in the following result. Theorem 3.2.
Let µ ∈ M + c a non-null measure such that (cid:104) µ , m (cid:105) < + ∞ , and set M := (cid:104) µ , m (cid:105) ∈ (0 , + ∞ ) , K := (cid:104) µ , m (cid:105) ∈ (0 , + ∞ ] . Let T gel := 1 /K ∈ [0 , + ∞ ) . Then Flory’s equation (3.1) has a unique solution ( µ t ) t ≥ on R + . It has the following properties.(1) We have M t = g ( l t ) , where l t = 1 for t ≤ T gel and, for t > T gel , l t is uniquelydefined by l t = e − t ( M − g ( l t )) , l t ∈ [0 , . Therefore t (cid:55)→ M t is continuous on [0 , + ∞ ) , constant on [0 , T gel ] , strictlydecreasing on [ T gel , + ∞ ) and analytic on R + \{ T gel } .(2) The function φ t ( x ) = xe t ( M − g ( x )) has a right inverse h t : [0 , → [0 , l t ] .The generating function g t of ( µ t ) t ≥ is given for t ≥ by g t ( x ) = g ( h t ( x )) . (3) Let m = inf supp µ ≥ . Then, when t → + ∞ , M t e m t → m µ ( { m } ) and for every (cid:15) > M t e ( m + (cid:15) ) t → + ∞ . (4) The second moment (cid:104) m , c t (cid:105) is finite on R + \{ T gel } and infinite at T gel . Remark 3.3. • Norris [24, Thm 2.8] has a proof of global uniqueness ofFlory’s equation (3.1) for slightly less general initial conditions ( µ such that (cid:104) µ , m (cid:105) < + ∞ ), but for a much more general model. • When m >
0, it was already observed (Proposition 5.3 in [10]) that themass decays (at least) exponentially fast (see also [8, 25, 31]).
Proof of Theorem 3.2.
The proof is very similar to (and actually easier than) thatof Theorem 2.2.(1) Arguing as in the proof of Lemma 2.4, we obtain easily that ( M t ) t ≥ ismonotone non-increasing and right-continuous. As in Lemma 2.5, if t (cid:55)→(cid:104) µ t , m (cid:105) is bounded on some interval [0 , T ], then M t = M for t ∈ [0 , T ] andtherefore ( µ t ) t ≥ is a solution of Smoluchowski’s equation (2.1) on [0 , T ].(2) Consider initial concentrations µ as in the statement, a solution ( µ t ) t ≥ toFlory’s equation and g t ( x ), x ∈ [0 , m µ t ( dm ). Then g t solves the PDE ∂g t ∂t = x ( g t − M ) ∂g t ∂x , ∀ t > , x ∈ [0 , , (3.2) OST-GELATION UNIQUENESS FOR COAGULATION EQUATIONS 19 the same as the one obtained for Smoluchowski’s equation before gelation.It may be solved using the method of characteristics. Indeed, the mapping φ t ( x ) = xe t ( M − g ( x )) = x + (cid:90) t ( M − g ( x )) φ s ( x ) d s, (3.3)has the following properties(d1) φ t (0) = 0, φ t (1) = 1.(d2) For all t ≥ φ (cid:48) t ( x ) = e t ( M − g ( x )) (1 − txg (cid:48) ( x )).(d3) For t ≤ T gel , φ t ( · ) is increasing; therefore, φ t ( x ) ∈ [0 ,
1] for all x ∈ [0 , φ t ( x ) = 1 if and only if x = 1(d4) For t > T gel , φ t ( · ) is increasing on [0 , m t ] and decreasing on [ m t , m t is the unique x ∈ (0 ,
1) such that φ (cid:48) t ( x ) = 0, i.e. such that txg (cid:48) ( x ) = 1.(d5) For t > T gel , φ t ( m t ) >
1, since φ t (1) = 1 and φ (cid:48) t (1) <
0. Therefore thereis a unique l t ∈ (0 , m t ) such that φ t ( l t ) = 1.(d6) For t > T gel , φ (cid:48) t ( l t ) (cid:54) = 0, since l t < m t . Figure 2. φ t before and after gelation.Setting l t := 1 for t ≤ T gel , φ t is thus a continuous bijection from [0 , l t ] to[0 , h t : [0 , (cid:55)→ [0 , l t ]. By using (3.2) and(3.3) and arguing as in part (i) and (ii) of the proof of Lemma 2.9, we cansee that the function u t ( x ) := g t ( φ t ( x )) − g ( x ) satisfies u t ( x ) = u ( x ) = 0for all t ≥ x ∈ [0 , l t ]. Therefore the only solution of the PDE (3.2) isgiven by g t ( x ) = g ( h t ( x )) , t ≥ , x ∈ [0 , . (3.4)Flory’s equation has thus a unique solution on R + , and its generating func-tion is g t .(3) We have seen in (d5) above that, for t > T gel , there is a unique l t ∈ [0 , φ t ( l t ) = 1. The relation φ t ( l t ) = 1 with l t ∈ [0 ,
1) is equivalent to l t = e − t ( M − g ( l t )) with l t ∈ [0 , t (cid:55)→ l t is analytic for t > T gel . A differentiation shows that dl t dt = − ( M − g ( l t )) l t − tg (cid:48) ( l t ) l t < , t > T gel , since g (cid:48) ( l t ) l t < g (cid:48) ( m t ) m t = 1 /t and g ( l t ) < g (1) = M . Let (cid:96) be the limitof l t as t ↓ T gel : then we obtain (cid:96) = e − T gel ( M − g ( (cid:96) )) , i.e. φ T gel ( (cid:96) ) = 1. By (d3)above, this is equivalent to (cid:96) = 1.(4) Since M t = g t (1) = g ( h t (1)) = g ( l t ), the properties of t (cid:55)→ M t = g ( l t )follow from those of t (cid:55)→ l t . Recall now that φ t ( l t ) = 1, that islog( l t ) = t ( g ( l t ) − . (3.5)If the limit l of l t as t → + ∞ were nonzero, then passing to the limit in thisequality would give log( l ) = −∞ . So l = 0 andlog l t ∼ − t. (3.6) • Assume m >
0. Now, obviously g ( x ) ≤ x m , solog( tg ( l t )) = log l t + log g ( l t ) ≤ log t + m log l t → −∞ . Hence tg ( l t ) → l t + t →
0. Hence l mt ∼ e − mt .Finally lim t → + ∞ M t e mt = lim t → + ∞ g ( l t ) l mt = mµ ( { m } )since by dominated convergence, g ( x ) x − m → mµ ( { m } ) when x → m (cid:48) > m , then g ( x ) x − m (cid:48) → + ∞ when x tends to 0, whencelim t → + ∞ M t e m (cid:48) t = lim t → + ∞ g ( l t ) l m (cid:48) t = + ∞ . • Assume now m = 0 and let (cid:15) >
0. By monotone convergence g ( x ) x − (cid:15) → + ∞ as x ↓
0, so using (3.6) we see that g ( l t ) e − (cid:15)t → + ∞ as t ↑ + ∞ ,which is the desired result.(5) Finally, (3.4) gives for x < t > T gel g (cid:48) t ( x ) = g (cid:48) ( h t ( x )) h (cid:48) t ( x ) = g (cid:48) ( h t ( x )) φ (cid:48) t ( h t ( x )) . When x ↑ h t ( x ) ↑ l t <
1, and φ (cid:48) t ( h t ( x )) → φ (cid:48) t ( l t ) (cid:54) = 0 by (d6) above. So (cid:104) µ t , m (cid:105) = g (cid:48) t (1) < + ∞ .(6) Existence of a solution of (3.1) follows arguing as in section 2.4. (cid:3) Corollary 3.4.
Let µ ∈ M + c such that (cid:104) µ , m (cid:105) < + ∞ and let ( µ St ) t ≥ and ( µ Ft ) t ≥ the solutions of (2.1) , respectively, (3.1) . Then • µ St ≡ µ Ft for all t ≤ T gel := 1 / (cid:104) µ , m (cid:105) ; • (cid:104) µ Ft , m (cid:105) < (cid:104) µ St , m (cid:105) for all t > T gel . OST-GELATION UNIQUENESS FOR COAGULATION EQUATIONS 21
Proof.
For all t ≤ T gel , (cid:104) µ Ft , m (cid:105) = (cid:104) µ F , m (cid:105) and therefore µ Ft solves (2.1), so that byuniqueness of Smoluchowski’s equation we have that µ St = µ Ft . For t > T gel we havethat (cid:104) µ Ft , m (cid:105) = g ( l t ) while (cid:104) µ St , m (cid:105) = g ( (cid:96) t ), where l t and (cid:96) t are defined respectivelyby l t = e − t ( M − g ( l t )) , l t ∈ [0 , (cid:96) t g (cid:48) ( (cid:96) t ) = 1 t . In points (d4) and (d5) of the proof of Theorem 3.2, we have shown that l t < m t where tm t g (cid:48) ( m t ) = 1, so that m t = (cid:96) t < l t . Hence (cid:104) µ Ft , m (cid:105) = g ( (cid:96) t ) < g ( l t ) = (cid:104) µ St , m (cid:105) . (cid:3) As anticipated, the mass decreases faster in Flory’s case than for Smoluchowski’sequation. In particular, in Flory’s case (cid:104) µ t , m (cid:105) becomes finite immediately aftergelation, the mass remaining however continuous (we can think that the big particles,which have the biggest influence on this second moment, disappear into the gel).Moreover, if inf supp µ > /t in Smoluchowski’s equation. Remark 3.5.
The mass in Flory’s equation may decrease slower if inf supp µ = 0.For instance, if µ (d m ) = e − m d m , then M t ∼ t − .4. The model with limited aggregation
We now turn to our main interest, namely Equation (1.2). We apply the sametechniques as above in a slightly more complicated setting. After giving all detailsin Smoluchowski’s case, we will give a shorter proof and focus on the differenceswith the proof of Theorem 2.2. As above, we can transform the system (1.2) intoa non-local PDE problem, which we are able to solve, thus obtaining existence anduniqueness to (1.2). More precisely, we consider the following system.
Definition 4.1.
Let c ( a, m ) ≥ , a ∈ N , m ∈ N ∗ . We say that a family ( c t ( a, m )) , t ≥ , a ∈ N , m ∈ N ∗ , is a solution of Smoluchowski’s equation (4.1) if • for every t > , (cid:82) t (cid:104) c s , a (cid:105) d s < + ∞ , • for all a ∈ N , m ∈ N ∗ and t > , c t ( a, m ) = c ( a, m )++ (cid:90) t a +1 (cid:88) a (cid:48) =1 m − (cid:88) m (cid:48) =1 a (cid:48) ( a + 2 − a (cid:48) ) c s ( a (cid:48) , m (cid:48) ) c s ( a + 2 − a (cid:48) , m − m (cid:48) ) d s − (cid:90) t (cid:88) a (cid:48) ≥ (cid:88) m (cid:48) ≥ aa (cid:48) c s ( a, m ) c s ( a (cid:48) , m (cid:48) ) d s, (4.1) • if (cid:104) c , a (cid:105) < + ∞ , then t (cid:55)→ (cid:104) c t , a (cid:105) is bounded in a right neighborhood of 0. Because of the interpretation of a as a variable counting the number of arms aparticle possesses, it is more natural to state (4.1) in the discrete setting, as in [3].In particular, since at each coagulation two arms are removed from the system, a non-integer initial number of arms would lead to an ill-defined dynamics. One couldhowever with no difficulty consider an initial distribution of masses on (0 , + ∞ ).It is easy to see that ( c t ) is a solution to this equation if and only if the function k t ( x, y ) := + ∞ (cid:88) a =1 + ∞ (cid:88) m =1 a c t ( a, m ) x a − y m , (4.2)defined for t ≥ y ∈ [0 ,
1] and x ∈ [0 , k t ( x, y ) = k ( x, y ) + (cid:90) t (cid:20) ( k s ( x, y ) − xA s ) ∂k s ∂x ( x, y ) − A s k s ( x, y ) (cid:21) d s,A t := k t (1 ,
1) = (cid:104) c t , a (cid:105) . (4.3)We may solve this PDE with the same techniques as above and obtain the followingresult. Theorem 4.2.
Consider initial concentrations c ( a, m ) ≥ , a ∈ N , m ∈ N ∗ suchthat (cid:104) c , (cid:105) < + ∞ , A := (cid:104) c , a (cid:105) ∈ (0 , + ∞ ] and with K := (cid:104) c , a ( a − (cid:105) ∈ [0 , + ∞ ] .Then K = + ∞ whenever A = + ∞ . Let T gel = K − A if A < K < + ∞ , K = + ∞ , + ∞ if K ≤ A < + ∞ . (4.4) Then equation (4.1) has a unique solution defined on R + . When T gel < + ∞ , thissolution enjoys the following properties.(1) The number of arms A t := (cid:104) c t , a (cid:105) is continuous, strictly decreasing, and forall t > A t ≤ A tA if A < + ∞ , A t ≤ t if A = + ∞ . (4.5) If we set α t = exp (cid:18)(cid:90) t A s d s (cid:19) , then α t is given by α t = 1 + A t for t < T gel and for t ≥ T gel α t = (cid:40) Γ − (1 + A T gel + t − T gel ) if A < + ∞ , Γ − (1 + t ) if A = + ∞ , (4.6) where Γ( x ) = 1 + A T gel + (cid:90) x A T gel drk ( H (1 /r )) , x ≥ A T gel , and H : [ G (0) , G (1)) (cid:55)→ [0 , is the right inverse of the increasing function G : [0 , (cid:55)→ [ G (0) , G (1)) , G ( x ) := x − k ( x, k (cid:48) ( x, , x ∈ [0 , , (4.7) OST-GELATION UNIQUENESS FOR COAGULATION EQUATIONS 23 with G (0) := G (0 + ) ≤ , and < G (1) := G (1 − ) = − A K if A < + ∞ A = + ∞ . (2) Let k be defined as in (4.2) , and A t = (cid:104) c t , a (cid:105) , α t = exp (cid:18)(cid:90) t A s d s (cid:19) , β t = (cid:90) t α s d s. (4.8) Consider φ t ( x, y ) := α t ( x − β t k ( x, y )) , t ≥ , x, y ∈ [0 , . Then • φ t ( · , attains its maximum at a point (cid:96) t such that φ t ( (cid:96) t ,
1) = 1 . For t ≤ T gel , (cid:96) t = 1 , and for t > T gel , < (cid:96) t < and ∂φ t ∂x ( (cid:96) t ,
1) = 0 . (4.9) In particular, for t > T gel , (cid:96) t is given by (cid:96) t = H (cid:18) α t (cid:19) , (4.10) where H is the right inverse of the function G defined above. • For every y ∈ [0 , , φ t ( · , y ) has a right inverse h t ( · , y ) : [0 , (cid:55)→ [0 , .(3) The generating function k t defined by (4.2) is given by k t ( x, y ) = 1 α t k ( h t ( x, y ) , y ) (4.11) for y ∈ [0 , , x ∈ [0 , . In particular, for t > α t A t = α t k t (1 ,
1) = k ( (cid:96) t , , A t = k ( (cid:96) t , (cid:82) t k ( (cid:96) s ,
1) d s . (4.12) (4) The second moment (cid:104) c t , a (cid:105) is finite on [0 , T gel ) , infinite on [ T gel , + ∞ ) . Proof.
The only major difference with respect to the proof of Theorem 2.2 isthe additional variable y in the generating function k t ( x, y ). However, the variable y plays the role of a parameter in the PDE (4.3), and this allows to adapt all abovetechniques. Proof of Theorem 4.2.
The case K ≤ A < + ∞ , for which T gel = + ∞ has alreadybeen treated in [3, Thm. 2], so that we can restrict here to the cases where T gel < + ∞ . When T gel >
0, Thm. 2 in [3] also shows that α t = 1 + A t on [0 , T gel ) (thishowever also requires that (cid:104) a , c t (cid:105) be bounded in a neighborhood of 0: see point 3of the proof of Lemma 2.9). (1) First, by setting u t ( x, y ) := α t k t ( φ t ( x, y ) , y ) − k ( x, y ), we can see, arguing asin points (i)-(ii) of the proof of Lemma 2.9, that for all y ∈ (0 ,
1] and t > (cid:96) t ( y ) < (cid:96) t ( y ) ∈ (0 ,
1] such that α t k t ( φ t ( x, y ) , y ) = k ( x, y ) , ∀ t ≥ , y ∈ (0 , , x ∈ [ (cid:96) t ( y ) , (cid:96) t ( y )] (4.13)and φ t ( · , y ) : [ (cid:96) t ( y ) , (cid:96) t ( y )] (cid:55)→ [0 ,
1] is a continuous bijection and has a contin-uous right inverse h t ( · , y ) : [0 , (cid:55)→ [ (cid:96) t ( y ) , (cid:96) t ( y )]. Figure 3. φ t ( · ,
1) before and after gelation. The dotted lines repre-sent what φ t may look like. The solid one is the actual φ t .(2) We denote for simplicity k t ( x ) := k t ( x, , φ t ( x ) := φ t ( x, , t ≥ , x ∈ [0 , . For y = 1, we set (cid:96) t (1) = (cid:96) t , i.e.1 = φ t ( (cid:96) t ) = α t ( (cid:96) t − β t k ( (cid:96) t )) , t ≥ . Arguing as in points (iv)-(v) of the proof of Lemma 2.9, we can see that (cid:96) t = 1 for all t ≤ T gel and (cid:96) t < t > T gel . Moreover, t (cid:55)→ (cid:96) t iscontinuous and monotone non-increasing. Since φ t is increasing on [0 , (cid:96) t ], φ (cid:48) t ( (cid:96) t ) ≥
0, i.e. β t ≤ k (cid:48) ( (cid:96) t ) , so that 1 = α t ( (cid:96) t − β t k ( (cid:96) t )) ≥ α t G ( (cid:96) t ) , (4.14)where we set G ( x ) := x − k ( x ) k (cid:48) ( x ) , x ∈ [0 , G (cid:48) ( x ) = 1 − ( k (cid:48) ( x )) − k ( x ) k (cid:48)(cid:48) ( x )( k (cid:48) ( x )) = k ( x ) k (cid:48)(cid:48) ( x )( k (cid:48) ( x )) > , since k is strictly convex (there is no gelation whenever k (cid:48)(cid:48) ≡ G (0) ≤ G (1) = 1 − A K if A < + ∞ , G (1) = 1 if A = + ∞ . OST-GELATION UNIQUENESS FOR COAGULATION EQUATIONS 25
Indeed, k (cid:48) (1) = K = (cid:104) c , a ( a − (cid:105) and, if k (1) = A = + ∞ , thenlim x ↑ k ( x ) k (cid:48) ( x ) = 0since, if lim inf x ↑ k ( x ) k (cid:48) ( x ) > ε >
0, then k (1) ≤ k (1 − δ ) e δ/ε < + ∞ , for some δ >
0, contradicting k (1) = + ∞ . In any case, G has an inverse H , and H (1 /x ) is defined for x ∈ [1 + A T gel , + ∞ ).(3) Computing (4.13) at ( x, y ) = ( (cid:96) t ,
1) we obtain k ( (cid:96) t ) = α t k t (1) = α t A t = d + α t d t . (4.15)Let us notice that φ t ( x ) = x + (cid:90) t (cid:18) A s φ s ( x ) − k ( x ) α s (cid:19) d s. Then by (4.15), analogously to (2.19) above,0 = d φ t ( (cid:96) t ) = (cid:18) A t φ t ( (cid:96) t ) − k ( (cid:96) t ) α t (cid:19) d t + φ (cid:48) t ( (cid:96) t ) d (cid:96) t = φ (cid:48) t ( (cid:96) t ) d (cid:96) t . In particular, for d (cid:96) t -a.e. t , φ (cid:48) t ( (cid:96) t ) = 0, i.e. β t = 1 /k (cid:48) ( (cid:96) t ), and therefore1 = α t ( (cid:96) t − β t k ( (cid:96) t )) = α t G ( (cid:96) t ) , d (cid:96) t − a . e . t. Then, by (4.14), we can write (note that H is well-defined on the consideredinterval) (cid:96) t ≤ H (cid:18) α t (cid:19) , ∀ t > T gel , (cid:96) t = H (cid:18) α t (cid:19) , d (cid:96) t − a . e . t. Now, by (4.15), setting Λ : ]1 + A T gel , + ∞ [ (cid:55)→ ]0 , z ) := k (cid:0) H (cid:0) z (cid:1)(cid:1) ,d + α t d t ≤ Λ( α t ) , ∀ t > T gel , d + α t d t = Λ( α t ) , d (cid:96) t − a . e . t. Since α t > A T gel for any t > T gel , we obtain that k ( (cid:96) t ) ≤ Λ( α t ) < t > T gel . In particular, d (cid:96) t is not identically equal to 0. Suppose that forsome t > T gel we have φ (cid:48) t ( (cid:96) t ) >
0. We set s := sup { r < t : φ (cid:48) r ( (cid:96) r ) = 0 } = max { r < t : φ (cid:48) r ( (cid:96) r ) = 0 } . Then for all r ∈ ] s, t [ we must have φ (cid:48) r ( (cid:96) r ) >
0. Then for all r ∈ ] s, t [ we have (cid:96) r = (cid:96) s . But, by definition of β , β r > β s = 1 k (cid:48) ( (cid:96) s ) = 1 k (cid:48) ( (cid:96) r )and this is a contradiction. Therefore for all t > T gel , we have ˙ α t = Λ( α t ) forall t > T gel and the only solution of this equation with α T gel = 1 + A T gel isgiven by (4.6). (4) In order to prove (4.12), let us note that by the preceding resultsd α t d t = α t A t = α t k t (1 ,
1) = k ( (cid:96) t , ,A t = dd t log α t = dd t log (cid:18) (cid:90) t k ( (cid:96) s ,
1) d s (cid:19) = k ( (cid:96) t , (cid:82) t k ( (cid:96) s ,
1) d s .
The rest of the proof follows the same line as that of Theorem 2.2. (cid:3) The modified version
Let us finally consider Flory’s version of the model with arms. As in the caseof Flory’s equation (3.1), we can consider only initial concentrations c such that A = (cid:104) c , a (cid:105) < + ∞ . Then, the equation we are interested in isdd t c t ( a, m ) = 12 a +1 (cid:88) a (cid:48) =1 m − (cid:88) m (cid:48) =1 a (cid:48) ( a + 2 − a (cid:48) ) c t ( a (cid:48) , m (cid:48) ) c t ( a + 2 − a (cid:48) , m − m (cid:48) ) − (cid:88) a (cid:48) ≥ (cid:88) m (cid:48) ≥ aa (cid:48) c t ( a, m ) c t ( a (cid:48) , m (cid:48) ) − (cid:32) A tA − (cid:88) a (cid:48) ,m (cid:48) ≥ a (cid:48) c t ( a (cid:48) , m (cid:48) ) (cid:33) ac t ( a, m ) . (5.1)With the same techniques as above, we can prove the following result. Theorem 5.1.
Consider initial concentrations c ( a, m ) ≥ , a ∈ N , m ∈ N ∗ suchthat A := (cid:104) c , a (cid:105) ∈ (0 , + ∞ ) and with K := (cid:104) c , a ( a − (cid:105) ∈ [0 , + ∞ ] . Let T gel bedefined as in (4.4) . Then equation (5.1) has a unique solution defined on R + . When T gel < + ∞ , this solution enjoys the following properties.(1) We have A t = 11 + tA k ( l t ) (5.2) where l t = 1 for t ≤ T gel and, for t > T gel , l t is uniquely defined by l t = t tA k ( l t ) , l t ∈ [0 , . Therefore t (cid:55)→ A t is continuous and strictly decreasing on [0 , + ∞ ) and ana-lytic on R + \{ T gel } .(2) The function φ t ( x, y ) = (1 + tA ) x − tk ( x, y ) has, for every y ∈ [0 , , a rightinverse h t ( · , y ) : [0 , → [0 , l t ] . The generating function k t defined in (4.2) is given for t ≥ by k t ( x, y ) = 11 + tA k ( h t ( x, y ) , y ) . (5.3) (3) The second moment (cid:104) a , c t (cid:105) is finite on R + \{ T gel } and infinite at T gel .Proof. The proof follows the same line of reasoning as the one of Theorem 3.2. First,for every y ∈ [0 , φ t ( · , y ), as defined in the statement, has the following properties: OST-GELATION UNIQUENESS FOR COAGULATION EQUATIONS 27 (i) φ t (0 , y ) ≤ φ t (1 , y ) ≥ φ t (1 ,
1) = 1;(ii) For t ≤ T gel , φ t ( · , y ) is increasing , and in particular, there are unique 0 ≤ l t ( y ) < l t ( y ) ≤ φ t ( l t ( y ) , y ) = 0 and φ t ( l t ( y ) , y ) = 1;(iii) For t > T gel , φ t ( · , y ) is increasing then decreasing for, and in particular,there are unique 0 ≤ l t ( y ) < l t ( y ) < φ t ( l t ( y ) , y ) = 0 and φ t ( l t ( y ) , y ) = 1. Figure 4. φ t ( · ,
1) before and after gelation.In any case, it is easy to check that for x ∈ [ l t ( y ) , l t ( y )],exp (cid:18)(cid:90) t A s d s (cid:19) k t ( φ t ( x, y ) , y ) = k ( x, y )where A t is defined by (5.2). Then, the properties above show that φ t ( · , y ) has aright inverse h t defined on [0 , h t in the previous equationshows that (5.3) holds. The other properties then follow easily. (cid:3) Limiting concentrations
We compute here some explicit formulas for the concentrations and their limitfor the two models above. In the standard Smoluchowski and Flory cases, particleskeep coagulating, and they all eventually disappear into the gel: c t ( m ) → m ≥
1. When the aggregations are limited, there may remain some particles in thesystem, since whenever a particle with no arms is created, it becomes inert, and soit will remain in the medium forever. In the following, we consider monodisperseinitial conditions, i.e. c ( a, m ) = µ ( a ) { m =1 } for a measure µ on N . We also denote ν ( m ) = ( m + 1) µ ( m + 1) . In [3], it is assumed that ν is a probability measure, what we do not require. Theresults of [3] can hence be recovered by taking A = 1 below. Now, note the twofollowing facts. • Equations (4.5) and (5.2) readily show that c ∞ ( a, m ) := lim t → + ∞ c t ( a, m ) = 0 , a ≥ , (6.1) that is, only particles with no arms remain in the medium (else, a coagulation“should” occur). • There is an arbitrary concentration of particles with no arms at time 0, andthey are the only particles with no arms and mass 1 which will still be in themedium in the final state. Hence, the limit concentrations c ∞ (0 ,
1) = c (0 , c ∞ (0 , m ) for m ≥ c ∞ ( m ) = 0for each m ≥
2. We thus rule out this trivial case by assuming that ν (0) > . (6.2)This is actually a technical assumption which is needed to apply Lagrange’s inversionformula in the proof of the following corollaries. We will relate our results to apopulation model known as the Galton-Watson process. For some basics on thistopic, see e.g. the classic book [2]. The formula providing the total progeny of theseprocesses was first obtained by Dwass in [7].6.1. Modified model.Corollary 6.1.
Let c t ( a, m ) be the solution to Flory’s equation with arms (5.1) andwith initial conditions c ( a, m ) = µ ( a ) { m =1 } with µ (1) > . • For all t ≥ , m ≥ , a ≥ , c t ( a, m ) = ( a + m − a ! m ! t m − (1 + tA ) a + m − ν ∗ m ( a + m − . • In particular, there are limiting concentrations c ∞ ( a, m ) = c ∞ ( m ) { a =0 } with c ∞ ( m ) = 1 m ( m − ν ∗ m ( m − . (6.3) Proof.
With the notation of Theorem 5.1, we have(1 + tA ) h t ( x, y ) − tyk ( h t ( x, y )) = x, k t ( x, y ) = 11 + tA yk ( h t ( x, y )) . Up to some obvious changes (just replace 1 + t by 1 + tA ), these are preciselythe equations solved in Section 3.2 of [3] under the assumption (6.2). Theorem 2and Corollary 2 therein hence give the desired result (with only 1 + t replaced by1 + tA ). (cid:3) If A = 1, which we may always assume up to a time-change, we observe asin [3] that 2( m − c ∞ (0 , m ) is the probability for a Galton-Watson process withreproduction law ν , started from two ancestors, to have total progeny m . ThisGalton-Watson process is (sub)critical when K := (cid:80) a ≥ a ( a − µ ( a ) ≤
1, that is,by Theorem 5.1, when there is no gelation, and supercritical when
K >
1. Denoteby p ν its extinction probability, i.e. the smallest root of k ( x ) = x , so p ν = 1 when OST-GELATION UNIQUENESS FOR COAGULATION EQUATIONS 29 K ≤ p ν < K >
1. Let us compute the mass at infinity, as in [3], bywriting M ∞ := (cid:88) m ≥ mc ∞ ( m ) = c ∞ (1) + (cid:88) m ≥ m − ν ∗ m ( m − c ∞ (1) + (cid:88) a ≥ ν ( a ) (cid:88) m ≥ a +2 m − ν ∗ m − ( m − − a )= c ∞ (1) + (cid:88) a ≥ ν ( a ) (cid:88) n ≥ a +1 n ν ∗ n ( n − − a ) . Now, the Lagrange inversion formula [30] shows that a + 1 n ν ∗ n ( n − − a )is precisely the coefficient of x n in the analytic expansion of φ ( x ) around 0, where φ is the unique solution to φ ( x ) = xk ( φ ( x )). Hence (cid:88) n ≥ a +1 n ν ∗ n ( n − − a ) = p ν , where p ν is defined above. Note also that c ∞ (1) = µ (0), so finally M ∞ = c ∞ (1) + (cid:88) a ≥ ν ( a ) 1 a + 1 p a +1 ν = (cid:88) a ≥ µ ( a ) p aν . (6.4)The mass at time 0 is M = (cid:80) µ ( a ), so when there is no gelation, p ν = 1 and nomass is lost in the gel. When there is gelation, p ν < M − M ∞ > M ∞ is also the probability that a Galton-Watson process, with reproduction law µ for the ancestor and ν for the others, hasa finite progeny.6.2. Non-modified model.Corollary 6.2.
Let c t ( a, m ) be the solution to Smoluchowski’s equation with arms (4.1) and with initial conditions c ( a, m ) = µ ( a ) { m =1 } with µ (1) > . • For all t ≥ , m ≥ , a ≥ , c t ( a, m ) = ( a + m − a ! m ! β m − t α at ν ∗ m ( a + m − where α t and β t are defined in Theorem 4.2. • In particular, there are limiting concentrations c ∞ ( a, m ) = c ∞ ( m ) { a =0 } with c ∞ ( m ) = 1 m ( m − β m − ∞ ν ∗ m ( m −
2) (6.5) where β ∞ is defined by β ∞ = 1 k (cid:48) ( c ) = ck ( c ) and c is the unique solution to k (cid:48) ( c ) = k ( c ) /c . Moreover, β ∞ = 1 whenthere is no gelation, and β ∞ > otherwise.Proof. As for Corollary 6.1, the proof of the formula for c t ( a, m ) is the same as in[3, Section 3.2], just replacing 1 + tA by α t and t by α t β t . So we just have to findthe limit of β t . First (4.6) shows that α t → + ∞ , hence, by (4.10), (cid:96) t → (cid:96) ∞ = H (0).Now, (4.9) gives β t = 1 /k (cid:48) ( (cid:96) t ), so β t tends to β ∞ = 1 k (cid:48) ( H (0))where by definition c := H (0) is the unique solution to k (cid:48) ( c ) = k ( c ) /c . Finally,when there is gelation, α t < t after gelation because of (4.6), so by (4.8), β ∞ > (cid:3) By a similar computation as above, we may also compute the mass at infinity inthis case and get M ∞ = (cid:88) a ≥ µ ( a ) c a where c is defined in the corollary. Note that c is the slope of the straight linepassing by 0 and tangent to the graph of k , so c > p ν . In particular, less mass islost than in Flory’s case.A final remark is that despite the striking resemblance between Formulas (6.5)and (6.3), the meaning of the factor β ∞ is unclear. A probabilistic interpretationusing the configuration model may explain its appearance. Acknowledgements
We thank Jean Bertoin for useful discussions and advice.
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