Absence of gelation and self-similar behavior for a coagulation-fragmentation equation
aa r X i v : . [ m a t h . A P ] J u l ABSENCE OF GELATION AND SELF-SIMILAR BEHAVIOR FORA COAGULATION-FRAGMENTATION EQUATION
PHILIPPE LAURENC¸ OT AND HENRY VAN ROESSEL
Abstract.
The dynamics of a coagulation-fragmentation equation with multiplicative coagulationkernel and critical singular fragmentation is studied. In contrast to the coagulation equation, it isproved that fragmentation prevents the occurrence of the gelation phenomenon and a mass-conservingsolution is constructed. The large time behavior of this solution is shown to be described by a self-similar solution. In addition, the second moment is finite for positive times whatever its initial value.The proof relies on the Laplace transform which maps the original equation to a first-order nonlinearhyperbolic equation with a singular source term. A precise study of this equation is then performedwith the method of characteristics. Introduction
Coagulation-fragmentation equations are mean-field models describing the growth of clusters chang-ing their sizes under the combined effects of (binary) merging and breakup. Denoting the size dis-tribution function of the particles with mass x > t > f = f ( t, x ) ≥
0, the coagulationequation with multiple fragmentation reads ∂ t f ( t, x ) = 12 Z x K ( x − y, y ) f ( t, x − y ) f ( t, y ) dy − Z ∞ K ( x, y ) f ( t, x ) f ( t, y ) dy − a ( x ) f ( t, x ) + Z ∞ x a ( y ) b ( x | y ) f ( t, y ) dy , (1.1)for ( t, x ) ∈ (0 , ∞ ) . In (1.1), K denotes the coagulation kernel which is a non-negative and symmetricfunction K ( x, y ) = K ( y, x ) ≥ x, y ) accounting for the likelihood of two particles with respectivemasses x and y to merge. Next, a ( x ) ≥ x > b ( x | y ) is the daughter distribution function describing the probability that the fragmentationof a particle with mass y produces a particle with mass x ∈ (0 , y ). Conservation of matter duringfragmentation events requires Z y xb ( x | y ) dx = y , y ∈ (0 , ∞ ) , (1.2) Date : September 1, 2018.1991
Mathematics Subject Classification.
Key words and phrases. coagulation, singular fragmentation, mass conservation, self-similarity, Laplace transform,characteristics. Ph. Laurenc¸ot & H. van Roessel and since the total mass of particles is preserved during coagulation events, the total mass is expectedto be constant throughout time evolution, that is, Z ∞ xf ( t, x ) dx = Z ∞ xf (0 , x ) dx , t > . (1.3)The total mass conservation (1.3) is actually a key issue in the analysis of coagulation-fragmentationequations as it might be infringed during time evolution. More precisely, in the absence of fragmen-tation ( a ≡ K split into two classes according totheir growth for large values of x and y : either K ( x, y ) ≤ κ (2 + x + y ) , ( x, y ) ∈ (0 , ∞ ) , (1.4)for some κ > mass-conserving andsatisfies (1.3). Or K ( x, y ) ≥ κ ( xy ) λ/ , ( x, y ) ∈ (0 , ∞ ) , (1.5)for some λ > κ > T gel , the so-called gelation time , such that Z ∞ xf ( t, x ) = Z ∞ xf (0 , x ) dx , t ∈ [0 , T gel ) and Z ∞ xf ( t, x ) < Z ∞ xf (0 , x ) dx , t > T gel . Alternatively, T gel := inf (cid:26) t > Z ∞ xf ( t, x ) dx < Z ∞ xf (0 , x ) dx (cid:27) . The fact that gelation occurs for coagulation kernels satisfying (1.5) was conjectured at the beginningof the eighties and supported by a few examples of explicit or closed-form solutions to (1.1) [20,21]. That it indeed takes place for such coagulation kernels and arbitrary initial data was showntwenty years later in [10]. Since fragmentation reduces the sizes of the clusters, it is rather expectedto counteract gelation and it has indeed been established that strong fragmentation prevents theoccurrence of gelation [5] but does not impede the occurrence of gelation if it is too weak [9,10,17,37].More specifically, for the following typical choices of coagulation kernel K and overall breakup rate aK ( x, y ) = κ (cid:0) x α y β + x β y α (cid:1) , ( x, y ) ∈ (0 , ∞ ) × (0 , ∞ ) , (1.6) a ( x ) = kx γ , x ∈ (0 , ∞ ) , with α ≤ β ≤ λ := α + β > γ > κ >
0, and k >
0, it is conjectured in [32, 37] thatmass-conserving solutions exist when γ > λ − γ < λ −
1, the criticalcase γ = λ − b . Except for the critical case γ = λ −
1, mathematical proofs of these conjecturesare provided in [5, 9, 10] for some specific classes of daughter distribution functions b . bsence of gelation and self-similarity for a coagulation-fragmentation equation 3 More generally, due to the possible breakdown of mass conservation, the analysis of the existenceof solutions to (1.1) follows two directions since the pioneering works by Melzak [27], McLeod [24–26],White [38], and Spouge [34]. On the one hand, several works have been devoted to the construction ofmass-conserving solutions to (1.1) for coagulation kernels satisfying (1.4) under various assumptionson the breakup rate a and the daughter distribution function b , see [1–3, 7, 9, 16, 18, 23, 27, 38] andthe references therein. On the other hand, weak solutions which need not satisfy (1.3) have beenconstructed for large classes of coagulation kernels K , breakup rate a , and daughter distributionfunction b , see [4, 8, 9, 11, 13, 14, 17, 31, 35]. Existence of mass-conserving solutions to (1.1) for coag-ulation kernels satisfying (1.5) with strong fragmentation has been established in [5, 9]. In addition,uniqueness of mass-conserving solutions has been investigated in [5, 15, 31, 36]. A survey of earlierresults and additional references may be found in [6, 19]. Let us mention here that a common featureof the above mentioned works is that the total number of particles n ( y ) resulting from the breakupof a cluster of size y > n ( y ) := Z y b ( x | y ) dx , (1.7)is assumed to be bounded or even constant and thus does not include the case where the breakup ofparticles could produce infinitely many particles. This possibility shows up in the class of breakuprates a and daughter distribution functions b satisfying (1.2) which are derived in [22] and given by a ( x ) = kx γ , b ( x | y ) = ( ω + 2) x ω y ω +1 , < x < y , (1.8)with k > γ ∈ R , and ω ∈ ( − , n ( y ) resulting from thebreakup of a particle of size y > ω ∈ ( − ,
0] and infinite if ω ∈ ( − , − λ = α + β and γ of K in (1.6) and a in (1.8) are related by γ = λ − α = β = 1 , κ = 12 , γ = 1 , ω = − , k > . (1.9)Note that the choice of ω in (1.9) corresponds to the production of an infinite number of particlesduring each fragmentation event. With this choice of parameters, equation (1.1) reads ∂ t f ( t, x ) = 12 Z x ( x − y ) yf ( t, x − y ) f ( t, y ) dy − Z ∞ xyf ( t, x ) f ( t, y ) dy − kxf ( t, x ) + k Z ∞ x yx f ( t, y ) dy . (1.10) Ph. Laurenc¸ot & H. van Roessel
At this point we realize that, introducing ν ( t, x ) := xf ( t, x ) and multiplying (1.10) by x , an alterna-tive formulation of (1.10) reads ∂ t ν ( t, x ) = x Z x ν ( t, x − y ) ν ( t, y ) dy − x Z ∞ ν ( t, x ) ν ( t, y ) dy − kxν ( t, x ) + k Z ∞ x ν ( t, y ) dy . (1.11)One advantage of this formulation is that it cancels the possible singularity of f ( t, x ) as x → ν ( t ) is expected to be integrable due to the boundedness of the total mass, itis unclear whether f ( t ) is integrable for positive times, even if the initial total number of particlesis finite. Indeed, recall that, for the fragmentation rate under consideration, an infinite number ofparticles is produced during each fragmentation event. Another advantage of this formulation andthe choice of the rate coefficients K , a , b is that taking the Laplace transform of (1.10) leads us tothe following partial differential equation ∂ t L ( t, s ) = ( L ( t,
0) + k − L ( t, s )) ∂ s L ( t, s ) + k L ( t, − L ( t, s ) s , t > , s > , (1.12)for the Laplace transform L ( t, s ) := Z ∞ ν ( t, x ) e − sx dx = Z ∞ xf ( t, x ) e − sx dx , ( t, s ) ∈ (0 , ∞ ) , of ν . Note that L ( t,
0) is the total mass at time t and does not depend on time in the absence ofgelation. Our aim is then to study the behavior of the solutions to (1.12) and thereby obtain someinformation on the behavior of solutions to (1.11) (and thus also on (1.10)). To this end, it turns outthat it is more appropriate to work with measure-valued solutions to (1.11). Let M + be the space ofnon-negative bounded measures on (0 , ∞ ) and fix an initial condition ν in satisfying ν in ∈ M + , Z ∞ ν in ( dx ) = 1 . (1.13)We may actually assume without loss of generality that ν in is a probability measure after a suitablerescaling. Weak solutions to (1.11) are then defined as follows: Definition 1.1.
Given an initial condition ν in satisfying (1.13) , a weak solution to (1.11) with initialcondition ν in is a weakly continuous map ν : [0 , ∞ ) → M + such that Z ∞ ϑ ( x ) ν ( t, dx ) = Z ∞ ϑ ( x ) ν in ( dx ) + Z t Z ∞ Z ∞ x [ ϑ ( x + y ) − ϑ ( x )] ν ( τ, dx ) ν ( τ, dy ) dτ + k Z t Z ∞ (cid:20)Z x ϑ ( y ) dy − xϑ ( x ) (cid:21) ν ( τ, dx ) dτ (1.14) for all t > and ϑ ∈ C ([0 , ∞ )) with compact support. We next denote the subset of non-negative bounded measures on (0 , ∞ ) with finite first momentby M +1 . We now state the main result: bsence of gelation and self-similarity for a coagulation-fragmentation equation 5 Theorem 1.2.
Let ν in be an initial condition satisfying (1.13) . There is a weak solution ν to (1.11) in the sense of Definition 1.1 which is mass-conserving for all times, that is, Z ∞ ν ( t, dx ) = 1 = Z ∞ ν in ( dx ) , t > . Moreover, ν ( t ) ∈ M +1 for all t > and Z ∞ xν ( t, dx ) ∼ e /k − t as t → ∞ . (1.15) Furthermore, introducing M ( t, x ) := Z x ν ( t, dy ) , ( t, x ) ∈ (0 , ∞ ) × (0 , ∞ ) , there is a probability measure ν ⋆ ∈ M + such that lim t →∞ M (cid:16) t, xt (cid:17) = M ⋆ ( x ) := Z x ν ⋆ ( dy ) for all x ∈ (0 , ∞ ) . (1.16)More information is actually available on ν ⋆ . Indeed, it follows from Proposition 5.1 that theLaplace transform L ⋆ of ν ⋆ is given by L ⋆ ( s ) := 1 + s − kW (cid:16) sk e (1+ s ) /k (cid:17) , s ≥ , (1.17)where W is the so-called Lambert W -function, that is, the inverse function of z ze z in (0 , ∞ ).Recalling that the gelation time is finite for all solutions to the coagulation equation (1.1) in theabsence of fragmentation ( k = 0), we deduce from Theorem 1.2 that adding fragmentation preventsthe occurrence of gelation whatever the value of k >
0. The value of the parameter k thus plays onlya minor role in that direction but it comes into play in the large time behavior as the self-similarprofile (1.17) depends explicitly on k . That a self-similar behavior for large times is plausible forrate coefficients K and a given by (1.6) and (1.8) and satisfying γ = α + β − ν (whichcorresponds to the second moment of f ) as t → ∞ gives a positive answer to a conjecture in [37],providing in addition an optimal rate of convergence to zero.As already mentioned, the proof of Theorem 1.2 relies on the study of the Laplace transform L of solutions ν to (1.11) which are related to solutions f of (1.10) by ν ( t, x ) = xf ( t, x ). A similartechnique has already been used for the coagulation equation with multiplicative kernel K ( x, y ) = xy and without fragmentation [28, 30, 33]. However, the fragmentation term complicates the analysis asit adds a singular reaction term in the first-order hyperbolic equation (1.12) solved by the Laplacetransform L . This additional difficulty is met again later on in the proof when the method ofcharacteristics is used. Indeed, in contrast to the case without fragmentation, it is no longer a singleordinary differential equation which shows up in the study of characteristics but a nonlinear systemof two ordinary differential equations with a singularity. To be more precise, the strategy to show Ph. Laurenc¸ot & H. van Roessel that (1.12) has a global solution L satisfying L ( t,
0) = 1 for all times t ≥ ∂ t ˜ L ( t, s ) = (1 + k − ˜ L ( t, s )) ∂ s ˜ L ( t, s ) + k − ˜ L ( t, s ) s , t > , s > , (which is nothing but (1.12) where we have replaced L ( t,
0) by 1) has a global solution ˜ L satisfying˜ L ( t,
0) = 1 for t ≥
0. Setting L = ˜ L obviously gives a global solution L to (1.12) satisfying L ( t,
0) = 1for t ≥
0. As already pointed out, the method of characteristics requires a detailed analysis of anonlinear system of two ordinary differential equations with a singularity which turns out to be ratherinvolved and is performed in Section 3.1. As a consequence of this analysis we obtain the existenceof a global solution L to (1.12) satisfying L ( t,
0) = 1 for t ≥ L to (1.12) iscompletely monotone for all positive times and thus the Laplace transform of a probability measure.Several auxiliary results on completely monotone functions are needed for this step. The existenceof a mass-conserving solution to (1.11) in the sense of Definition 1.1 results from the outcome ofSections 3 and 4. Additional information can be retrieved from the detailed study of L performed inSections 3 and 4. This allows us to identify the large time behavior of L in Section 5 as well as thebehavior of ∂ s L ( t, s ) as s → Alternative representation
Let ν be a weak solution to (1.11) in the sense of Definition 1.1. Introducing its Laplace transform L ( t, s ) := Z ∞ e − sx ν ( t, dx ) , ( t, s ) ∈ [0 , ∞ ) × (0 , ∞ ) , and observing that x e − sx is bounded and continuous for s >
0, we infer from (1.14) that L solves ∂ t L ( t, s ) = ( L ( t,
0) + k − L ( t, s )) ∂ s L ( t, s ) + k L ( t, − L ( t, s ) s , t > , s > , (2.1) L (0 , s ) = L ( s ) , s > , (2.2)where L ( s ) := Z ∞ e − sx ν in ( dx ) , s > . (2.3)Introducing the characteristic equation dSdt ( t ) = L ( t, S ( t )) − L ( t, − k , (2.4)we infer from (2.1) that ddt L ( t, S ( t )) = k L ( t, − L ( t, S ( t )) S ( t ) . (2.5) bsence of gelation and self-similarity for a coagulation-fragmentation equation 7 Consequently, t ( S ( t ) , L ( t, S ( t )) solves the differential system (2.4)-(2.5) which has a singularitywhen S ( t ) vanishes and is not closed as it features the yet unknown time dependent function L ( t, L ( t,
0) = 1 and thedifferential system (2.4)-(2.5) is closed and can in principle be solved.3.
Well-posedness
We first list some useful properties of the Laplace transform L of ν in defined in (2.3). Owing to(1.11), L ∈ C ([0 , ∞ )) ∩ C ∞ ((0 , ∞ )) and satisfies L ′ ( s ) = − Z ∞ xe − sx ν in ( dx ) and 0 < L ( s ) < , s > , (3.1a) sL ′ ( s ) + 1 − L ( s ) = Z ∞ ( e sx − − sx ) e − sx ν in ( dx ) > , s > , (3.1b)the second statement being a consequence of the elementary inequality e sx ≥ sx for x > s >
0. For further use, we also define L ( s ) := L ( s ) − s < L ′ ( s ) = sL ′ ( s ) + 1 − L ( s ) s > , s > , (3.2)the positivity properties of − L and L ′ being a consequence of (3.1a) and (3.1b).3.1. An auxiliary differential system.
According to the previous discussion, we focus in thissection on the following initial value problem: given s > d Σ dt ( t ) = ℓ ( t ) − − k , (3.3) dℓdt ( t ) = k − ℓ ( t )Σ( t ) , (3.4)(Σ , ℓ )(0) = ( s, L ( s )) . (3.5)We infer from the Cauchy-Lipschitz theorem that there is a unique maximal solution (Σ , ℓ )( · , s ) ∈ C ([0 , T ( s )); R ) to (3.3)-(3.5) such thatΣ( t, s ) > , t ∈ [0 , T ( s )) . (3.6)In addition, T ( s ) < ∞ ⇐⇒ lim t → T ( s ) Σ( t, s ) = 0 or lim t → T ( s ) (Σ( t, s ) + | ℓ ( t, s ) | ) = ∞ . (3.7)Since L ( s ) ∈ (0 ,
1) by (3.1a), a first consequence of (3.4) and the comparison principle is that ℓ ( t, s ) < t ∈ [0 , T ( s )). This fact and (3.6) ensure that the right hand side of (3.4) is thenpositive, hence ∂ t ℓ ( t, s ) > ≤ ℓ ( t, s ) < , t ∈ [0 , T ( s )) . (3.8)We then deduce from (3.3), (3.8), and the positivity of k that ∂ t Σ( t, s ) < − k < t, s ) ≤ s , t ∈ [0 , T ( s )) . (3.9) Ph. Laurenc¸ot & H. van Roessel
Two interesting consequences can be drawn from the estimates (3.8) and (3.9): they clearly excludethe occurrence of finite time blowup and imply that Σ( · , s ) vanishes at a finite time. Recalling (3.7),we conclude that T ( s ) < ∞ and lim t → T ( s ) Σ( t, s ) = 0 . In fact, Σ( · , s ), ℓ ( · , s ) and T ( s ) can be computed explicitly as we show now. Owing to (3.6) and(3.8), the function ln Σ − ln (1 − ℓ ) + ( ℓ/k ) is well-defined on [0 , T ( s )) and we infer from (3.3) and(3.4) that ddt (cid:18) ln Σ − ln (1 − ℓ ) + ℓk (cid:19) ( t, s ) = 0 , t ∈ [0 , T ( s ) . Therefore Σ( t, s ) = s − ℓ ( t, s )1 − L ( s ) e ( L ( s ) − ℓ ( t,s )) /k , t ∈ [0 , T ( s )) , (3.10a)or, alternatively, 1 − ℓ ( t, s )Σ( t, s ) = 1 − L ( s ) s e ( ℓ ( t,s ) − L ( s )) /k , t ∈ [0 , T ( s )) . (3.10b)Inserting (3.10b) in (3.4) gives dℓdt ( t, s ) = k − L ( s ) s e − L ( s ) /k e ℓ ( t,s ) /k , t ∈ [0 , T ( s )) , (3.11)whence, after integration, ℓ ( t, s ) = L ( s ) − k ln (1 + tL ( s )) , t ∈ [0 , T ( s )) , (3.12)the function L being defined in (3.2). We next combine (3.10a) and (3.12) to obtainΣ( t, s ) = − tL ( s ) L ( s ) [1 − L ( s ) + k ln (1 + tL ( s ))] , t ∈ [0 , T ( s )) . (3.13)The first term of the right hand side of (3.13) vanishes when t = − /L ( s ) = s/ (1 − L ( s )) while thesecond term is a decreasing function of time (recall that L ( s ) <
1) and ranges in ( −∞ , − L ( s )]when t ranges in [0 , − /L ( s )). It thus vanishes only once in [0 , − /L ( s )) and since we have alreadyexcluded finite time blowup, we conclude that T ( s ) is the unique zero in [0 , − /L ( s )) of the secondterm of the right hand side of (3.13), that is, T ( s ) solves1 − L ( s ) + k ln (1 + T ( s ) L ( s )) = 0 , (3.14a)or, alternatively, T ( s ) = s − L ( s ) (cid:0) − e ( L ( s ) − /k (cid:1) < s − L ( s ) . (3.14b)The last bound implies in particular that s (1 + tL ( s )) = s + ( L ( s ) − t > se ( L ( s ) − /k > , t ∈ [0 , T ( s )) . (3.15)Recalling that the differential system (3.5)-(3.6) features a singularity as t → T ( s ), let us investi-gate further the behavior of (Σ , ℓ )( t, s ) as t → T ( s ). bsence of gelation and self-similarity for a coagulation-fragmentation equation 9 Lemma 3.1.
The functions Σ( · , s ) and ℓ ( · , s ) both belong to C ([0 , T ( s )]) with Σ( T ( s ) , s ) = 0 , ℓ ( T ( s ) , s ) = 1 ,∂ t Σ( T ( s ) , s ) = − k , ∂ t ℓ ( T ( s ) , s ) = − kL ( s ) e ( L ( s ) − /k . Proof.
The behavior of Σ( t, s ) and ℓ ( t, s ) as t → T ( s ) is a straightforward consequence of (3.12),(3.13), and (3.14a). Next, it readily follows from (3.3) thatlim t → T ( s ) ∂ t Σ( t, s ) = − k . Consequently, Σ( · , s ) ∈ C ([0 , T ( s )]) and ℓ ( · , s ) shares the same regularity thanks to a similar argu-ment relying on (3.11). (cid:3) We now study the behavior of T ( s ) as a function of s > Lemma 3.2.
The function T is an increasing C ∞ -smooth diffeomorphism from (0 , ∞ ) onto (0 , ∞ ) and enjoys the following properties: lim s → T ( s ) = 0 and T ( s ) ∼ sk as s → , (3.16a)lim s →∞ T ( s ) = ∞ and T ( s ) ∼ (1 − e − /k ) s as s → ∞ . (3.16b) Proof.
The smoothness of T readily follows from that of L and (3.1a) by (3.14b) while (3.16a) and(3.16b) are consequences of (3.14b) and the properties of L . To establish the monotonicity of T , wedifferentiate (3.14a) with respect to s and find − L ′ ( s ) + k L ′ ( s ) T ( s ) + L ( s ) T ′ ( s )1 + L ( s ) T ( s ) = 0 . Since 1 + L ( s ) T ( s ) = e ( L ( s ) − /k by (3.14a), we obtain − L ′ ( s ) e ( L ( s ) − /k + k (1 + L ′ ( s ) T ( s ) + L ( s ) T ′ ( s )) = 0 , and then − kL ( s ) T ′ ( s ) = − L ′ ( s ) e ( L ( s ) − /k + k (1 + L ′ ( s ) T ( s )) . Since − L ′ and L ′ are both positive by (3.1a) and (3.2), we conclude that T ′ ( s ) > , s > , which, together with (3.16a) and (3.16b), implies that T is an increasing C ∞ -smooth diffeomorphismfrom (0 , ∞ ) onto (0 , ∞ ) and completes the proof. (cid:3) Denoting the inverse of T by T − , we deduce from (3.14a) that1 = L ( T − ( t )) − k ln (1 + tL ( T − ( t )) , t > . (3.17)We next turn to the properties of Σ with respect to the variable s and establish the followingmonotonicity result: Ph. Laurenc¸ot & H. van Roessel
Lemma 3.3.
Let t > . The function Σ( t, · ) is an increasing C ∞ -smooth diffeomorphism from ( T − ( t ) , ∞ ) onto (0 , ∞ ) and satisfies ∂ s Σ( t, s ) > for s ∈ ( T − ( t ) , ∞ ) and Σ( t, s ) ∼ s as s → ∞ . (3.18) Proof.
Let s > t ∈ (0 , T ( s )). We differentiate (3.13) with respect to s and obtain ∂ s Σ( t, s ) = − tL ( s ) L ( s ) (cid:20) − L ′ ( s ) + kt L ′ ( s )1 + tL ( s ) (cid:21) + L ′ ( s ) L ( s ) (1 − L ( s ) + k ln (1 + tL ( s )))= L ′ ( s ) L ( s ) + tL ′ ( s ) − kt L ′ ( s ) L ( s ) − sL ′ ( s ) L ( s ) + k L ′ ( s ) L ( s ) ln (1 + tL ( s )) ,∂ s Σ( t, s ) = 1 + tL ′ ( s ) + k L ′ ( s ) L ( s ) [ln (1 + tL ( s )) − tL ( s )] . (3.19)Differentiating (3.19) with respect to t gives ∂ t ∂ s Σ( t, s ) = L ′ ( s ) + k L ′ ( s ) L ( s ) (cid:20) L ( s )1 + tL ( s ) − L ( s ) (cid:21) = L ′ ( s ) − kt L ′ ( s )1 + tL ( s )) < t ∈ [0 , T ( s )), the negativity of the right hand side of the above inequality being a consequence of(3.1a), (3.2), and (3.15). Therefore, for all t ∈ [0 , T ( s )), ∂ s Σ( t, s ) > τ ( s ) := lim t → T ( s ) ∂ s Σ( t, s ) , (3.20)where τ ( s ) = 1 + T ( s ) L ′ ( s ) + k L ′ ( s ) L ( s ) [ln (1 + T ( s ) L ( s )) − L ( s ) T ( s )] . Since ln (1 + T ( s ) L ( s )) = L ( s ) − k = sL ( s ) k by (3.14a), we realize that τ ( s ) =1 + T ( s ) L ′ ( s ) + k L ′ ( s ) L ( s ) h sk − T ( s ) i = L ′ ( s ) (cid:18) T ( s ) − s − L ( s ) (cid:19) − k L ′ ( s ) L ( s ) T ( s ) > , due to (3.1a), (3.2), and (3.14b). Recalling (3.20), we have shown that ∂ s Σ( t, s ) > t ∈ (0 , T ( s ))and s > s ∈ ( T − ( t ) , ∞ ) and t >
0. Consequently, for each t >
0, Σ( t, · ) is anincreasing C ∞ -smooth diffeomorphism from ( T − ( t ) , ∞ ) onto its range which is nothing but (0 , ∞ ) bsence of gelation and self-similarity for a coagulation-fragmentation equation 11 since Σ( t, s ) ∼ s as s → ∞ by (3.13) and Σ( t, s ) → s → T − ( t ) by Lemma 3.1. The proof ofLemma 3.3 is then complete. (cid:3) For t >
0, we denote the inverse of Σ( t, · ) by ζ ( t, · ) and observe that it is a C ∞ -smooth increasingfunction from (0 , ∞ ) onto ( T − ( t ) , ∞ ). SinceΣ( t, ζ ( t, s )) = s for ( t, s ) ∈ (0 , ∞ ) , (3.21)we infer from (3.18), (3.21), and the implicit function theorem that ζ ∈ C ∞ ((0 , ∞ ) ) with ∂ s ζ ( t, s ) = 1 ∂ s Σ( t, ζ ( t, s )) and ∂ t ζ ( t, s ) = − ∂ t Σ( t, ζ ( t, s )) ∂ s Σ( t, ζ ( t, s )) , ( t, s ) ∈ (0 , ∞ ) . (3.22)We end up this section with the differentiability of Σ( t, · ) and ℓ ( t, · ) at s = T − ( t ). Lemma 3.4.
Let t > . Both Σ( t, · ) and ℓ ( t, · ) belong to C ([ T − ( t ) , ∞ )) .Proof. Since T − ( t ) > L , L ∈ C ((0 , ∞ )), we infer from (3.19) that ∂ s Σ( t, s )has a limit as s → T − ( t ) and thus that Σ( t, · ) ∈ C ([ T − ( t ) , ∞ ). Similarly, by (3.12), ∂ s ℓ ( t, s ) = L ′ ( s ) − kt L ′ ( s )1 + tL ( s ) , which has a limit as s → T − ( t ) as L and L belong to C ((0 , ∞ )). (cid:3) Existence of a solution to (2.1) - (2.2) . After this preparation, we are in a position to showthe existence of a solution to (2.1)-(2.2). As expected from the analysis performed in Section 2, weset L ( t, s ) := ℓ ( t, ζ ( t, s )) , ( t, s ) ∈ (0 , ∞ ) , (3.23)and aim at showing that L solves (2.1)-(2.2) as well as identifying L ( t,
0) for all t > ζ , (3.12), (3.17), and (3.23) entail that, for t > L ( t,
0) = lim s → L ( t, s ) = lim s → ℓ ( t, ζ ( t, s )) = lim s → T − ( t ) ℓ ( t, s )= L ( T − ( t )) − k ln (1 + tL ( T − ( t ))) = 1 ,L ( t,
0) = 1 , t > . (3.24)On the other hand, it follows from (3.3), (3.4), (3.22), and (3.23) that, for ( t, s ) ∈ (0 , ∞ ) , ∂ t L ( t, s ) = k − L ( t, s ) s + ∂ s ℓ ( t, ζ ( t, s )) ∂ t ζ ( t, s )= k − L ( t, s ) s − ∂ s ℓ ( t, ζ ( t, s )) ∂ t Σ( t, ζ ( t, s )) ∂ s Σ( t, ζ ( t, s ))= k − L ( t, s ) s − ∂ s L ( t, s ) ( L ( t, s ) − k − , Ph. Laurenc¸ot & H. van Roessel whence, thanks to (3.24), ∂ t L ( t, s ) = ∂ s L ( t, s ) ( L ( t,
0) + k − L ( t, s )) + k L ( t, − L ( t, s ) s , ( t, s ) ∈ (0 , ∞ ) . Finally, the properties of Σ, ζ , and (3.5) imply that, for s > t → L ( t, s ) = lim t → ℓ ( t, ζ ( t, s )) = ℓ (0 , s ) = L ( s ) . We have thus shown that the function L defined in (3.23) solves (2.1)-(2.2) and enjoys the additionalproperty (3.24). To obtain a solution to the coagulation-fragmentation equation (1.10) it remainsto show that, for all t > L ( t, · ) is the Laplace transform of a non-negative bounded measure oralternatively that it is completely monotone. This will be the aim of Section 4.4. Complete monotonicity
Proposition 4.1.
For all t > , the function L ( t, · ) defined in (3.23) is completely monotone. We first recall two important criteria guaranteeing complete monotonicity, see [12, Chapter XIII.4,Criterion 1 & Criterion 2] for instance.
Lemma 4.2. (1) If ϕ and ψ are completely monotone functions then their product ϕψ is also acompletely monotone function. (2) If ϕ is a completely monotone function and ψ is a non-negative function with a completelymonotone derivative, then ϕ ◦ ψ is a completely monotone function. In particular, the complete monotonicity of r
7→ − ln r and Lemma 4.2 (2) have the followingconsequence. Lemma 4.3.
Let I be an open interval of R and g : I (0 , be a C ∞ -smooth function such that g ′ is completely monotone. Then − ln g is completely monotone. Let t >
0. Recalling that L ( t, · ) is given by L ( t, s ) = ℓ ( t, ζ ( t, s )) , s > , the proof of its complete monotonicity is performed in two steps. More precisely, we prove that ℓ ( t, · )and the derivative of ζ ( t, · ) are completely monotone. The complete monotonicity of L ( t, · ) is then aconsequence of Lemma 4.2 (2) and the non-negativity of ζ ( t, · ). To prove the complete monotonicityof ℓ ( t, · ) we need the following result: Lemma 4.4.
Let g : (0 , ∞ ) (0 , be a completely monotone function satisfying g (0) = 1 . Setting G ( x ) := ( g ( x ) − /x for x ∈ (0 , ∞ ) , its derivative G ′ is completely monotone.Proof. We first note that G ′ ( x ) = xg ′ ( x ) − g ( x ) + 1 x = g ′ ( x ) − G ( x ) x , x ∈ (0 , ∞ ) . (4.1) bsence of gelation and self-similarity for a coagulation-fragmentation equation 13 Step 1:
We first prove by induction that, for n ≥ G ( n +1) ( x ) = g ( n +1) ( x ) − ( n + 1) G ( n ) ( x ) x , x ∈ (0 , ∞ ) , (4.2)with the obvious notation G (0) = G . Indeed, (4.2) is clearly true for n = 0 by (4.1). Assume nextthat (4.2) is true for some n ≥
0. Differentiating the corresponding identity gives, for x > G ( n +2) ( x ) = g ( n +2) ( x ) − ( n + 1) G ( n +1) ( x ) x − g ( n +1) ( x ) − ( n + 1) G ( n ) ( x ) x We next use (4.2) for n and find G ( n +2) ( x ) = g ( n +2) ( x ) − ( n + 1) G ( n +1) ( x ) x − G ( n +1) ( x ) x , whence (4.2) for n + 1. Step 2:
We next prove by induction that, for n ≥ G ( n ) ( x ) = n ! x n +1 " ( − n ( g ( x ) −
1) + n X j =1 ( − n − j x j j ! g ( j ) ( x ) , x ∈ (0 , ∞ ) . (4.3)Indeed, the identity (4.3) is true for n = 1 by (4.1). Assume next that G ( n ) is given by (4.3) for some n ≥
1. It then follows from (4.2) and (4.3) that, for x > G ( n +1) ( x ) = g ( n +1) ( x ) x − ( n + 1)! x n +2 " ( − n ( g ( x ) −
1) + n X j =1 ( − n − j x j j ! g ( j ) ( x ) = ( n + 1)! x n +2 " ( − n +1 ( g ( x ) −
1) + n X j =1 ( − n +1 − j x j j ! g ( j ) ( x ) + x n +1 ( n + 1)! g ( n +1) ( x ) . Thus, G ( n +1) is also given by (4.3). Step 3:
Define h n ( x ) := x n +1 G ( n ) ( x ) /n ! for n ≥ x >
0. Thanks to (4.3), h n ( x ) = ( − n ( g ( x ) −
1) + n X j =1 ( − n − j x j j ! g ( j ) ( x ) , x ∈ (0 , ∞ ) . Since g (0) = 1, we have h n (0) = 0 and, for x > h ′ n ( x ) =( − n g ′ ( x ) + n X j =1 ( − n − j x j j ! g ( j +1) ( x ) + n X j =1 ( − n − j x j − ( j − g ( j ) ( x )= n +1 X j =1 ( − n +1 − j x j − ( j − g ( j ) ( x ) + n X j =1 ( − n − j x j − ( j − g ( j ) ( x )= x n n ! g ( n +1) ( x ) . Ph. Laurenc¸ot & H. van Roessel
Since g is completely monotone, the previous identity implies that ( − n +1 h ′ n ≥
0. Recalling that h n (0) = 0, we have thus shown that ( − n +1 h n ≥ − n +1 G ( n ) ≥ G ′ . (cid:3) Lemma 4.5.
For each t > , ℓ ( t, · ) is completely monotone in ( T − ( t ) , ∞ ) .Proof. Fix t > ℓ ( t, · ) is given by ℓ ( t, s ) = L ( s ) − k ln (1 + tL ( s )) , s ∈ ( T − ( t ) , , with L ( s ) := ( L ( s ) − /s for s >
0. Since L is completely monotone with L (0) = 1, we infer fromLemma 4.4 that the derivative of the function L is completely monotone in (0 , ∞ ). Then, so is thederivative of 1 + tL and s tL ( s ) ranges in (0 ,
1) when s ∈ ( T − ( t ) , ∞ ) according to (3.15).We are thus in a position to apply Lemma 4.3 and conclude that s
7→ − ln (1 + tL ( s )) is completelymonotone in ( T − ( t ) , ∞ ). Since k > L is completely monotone in (0 , ∞ ), we conclude that ℓ ( t, · ) is completely monotone in ( T − ( t ) , ∞ ). (cid:3) We next turn to ζ ( t, · ) and first establish the following auxiliary result. Lemma 4.6.
Let g be a non-negative function in C ∞ (0 , ∞ ) such that g ′ is completely monotone and g ′ < . Then the function (id − g ) − has a completely monotone derivative.Proof. For the sake of completeness, we give a sketch of the proof which is actually outlined in [28,p. 1209]. We set σ ∞ := (id − g ) − and define by induction a sequence of functions ( σ n ) n ≥ as follows: σ ( s ) := s and σ n +1 ( s ) := s + g ( σ n ( s )) , s > , n ≥ . (4.4)On the one hand, thanks to the properties of g (and in particular the bounds 0 ≤ g ′ < σ n is well-defined and non-negative for all n ≥ C ∞ (0 , ∞ ). In addition, itsatisfies σ ( s ) = s ≤ σ n ( s ) ≤ σ n +1 ( s ) ≤ σ ∞ ( s ) , s > , n ≥ . (4.5)Now, fix s > s > s . Owing to the property g ′ <
1, there is δ ∈ (0 ,
1) such that g ′ ( s ) ≤ δ foreach s ∈ [ s , σ ∞ ( s )]. Since σ n ( s ) ∈ [ s , σ ∞ ( s )] for n ≥ s ∈ [ s , s ] by (4.5), we obtain0 ≤ σ ∞ ( s ) − σ n +1 ( s ) = g ( σ ∞ ( s )) − g ( σ n ( s )) ≤ δ ( σ ∞ ( s ) − σ n ( s )) . This estimate readily implies that( σ n ) n ≥ converges uniformly towards σ ∞ on compact subsets of (0 , ∞ ) . (4.6)On the other hand, it follows from (4.4) by induction that σ n is completely monotone for every n ≥ . (4.7)Indeed, σ = id is clearly completely monotone and, if σ n is assumed to be completely monotone,the complete monotonicity of σ n +1 follows from (4.4) and that of σ n and g ′ with the help of [12,Chapter XIII.4, Criterion 2].The assertion of Lemma 4.6 then follows from (4.6) and (4.7) by [12, Chapter XIII.1, Theorem 2& Chapter XIII.4, Theorem 1]. (cid:3) bsence of gelation and self-similarity for a coagulation-fragmentation equation 15 Lemma 4.7.
For each t > , ∂ s ζ ( t, · ) is completely monotone in (0 , ∞ ) .Proof. Fix t >
0. IntroducingΦ( s ) := s − Σ( t, s ) = (1 − L ( s )) t + k tL ( s ) L ( s ) ln (1 + tL ( s )) , s ∈ ( T − ( t ) , ∞ ) , the formula (3.13) also reads Σ( t, s ) = s − Φ( s ) for s ∈ ( T − ( t ) , ∞ ) from which we deduce that s = ζ ( t, s ) − Φ( ζ ( t, s )) , s > . (4.8)Therefore, ζ ( t, · ) satisfies a functional identity of the form required to apply Lemma 4.6. To go on,we have to show that Φ ′ is completely monotone. To this end, we compute Φ ′ and findΦ ′ ( s ) = − tL ′ ( s ) + k L ′ ( s ) L ( s ) [ tL ( s ) − ln (1 + tL ( s ))] , s ∈ ( T − ( t ) , ∞ ) . Given θ ∈ [0 , t ], the monotonicity of T − ensures that T − ( θ ) ≤ T − ( t ) so that 1 + θL > T − ( t ) , ∞ ) by (3.15). In addition, 1 + θL has a completely monotone derivative since L ′ iscompletely monotone by Lemma 4.4 which, together with the complete monotonicity of z /z and Lemma 4.1 (2) entails the complete monotonicity of 1 / (1 + θL ) in ( T − ( t ) , ∞ ) for all θ ∈ [0 , t ].Observing that tL ( s ) − ln (1 + tL ( s )) L ( s ) = Z t θ θL ( s ) dθ , s ∈ ( T − ( t ) , ∞ ) , we infer from [29, Theorem 4] that tL − ln (1 + tL ) L is completely monotone in ( T − ( t ) , ∞ ) . Using now Lemma 4.1 (1) along with the positivity of k and the complete monotonicity of L ′ and − L ′ entails that Φ ′ is completely monotone in ( T − ( t ) , ∞ ). Furthermore, recalling the definition ofΦ, there holds Φ ′ = 1 − ∂ s Σ( t, · ) < ∈ C ∞ (( T − ( t ) , ∞ )) has a completely monotone derivativeΦ ′ satisfying Φ ′ < ζ ( t, · ) solves (4.8). Lemma 4.6 then ensures that ∂ s ζ ( t, · ) is completelymonotone in (0 , ∞ ). (cid:3) Proof of Proposition 4.1.
Fix t >
0. The complete monotonicity of L ( t, · ) is a straightforward con-sequence of the non-negativity of ζ ( t, · ), Lemma 4.1 (2), Lemma 4.5, and Lemma 4.7. (cid:3) Now, for each t > L ( t, · ) is completely monotone in (0 , ∞ ) with L ( t,
0) = 1 by Proposition 4.1 and(3.24) so that it is the Laplace transform of a probability measure ν ( t ) ∈ M + by [12, Chapter XIII.4,Theorem 1]. In addition, it follows from the time continuity of L and [12, Chapter XIII.1, Theorem 2]that the map ν : [0 , ∞ ) → M + is weakly continuous. We finally argue as in [28, Section 2] toshow that ν satisfies (1.14) for all C -smooth functions ϑ with compact support. An additionalapproximation argument allows us to extend the validity of (1.14) to all continuous functions ϑ withcompact support and complete the proof of the first statement of Theorem 1.2. Ph. Laurenc¸ot & H. van Roessel Large time behavior
We now aim at investigating the behavior of L ( t, ts ) as t → ∞ for any given s >
0. Morespecifically, we prove the following convergence result.
Proposition 5.1.
For all s > , there holds lim t →∞ L ( t, ts ) = 1 + s − kW (cid:16) sk e (1+ s ) /k (cid:17) , (5.1) where we recall that W is the Lambert W -function, that is, the inverse function of z ze z in (0 , ∞ ) .Proof. We fix s > η ( t ) := ζ ( t, ts ) and µ ( t ) := − tL ( η ( t )) , t > . (5.2)Since η ( t ) = ζ ( t, ts ) ≥ T − ( t ), Lemma 3.2 ensures thatlim t →∞ η ( t ) = ∞ . (5.3)Next, for σ > T − ( t ), we infer from the properties of L and (3.14b) that1 > tL ( σ ) > L ( σ ) T ( σ ) = e ( L ( σ ) − /k . Taking σ = η ( t ) in the above estimate gives1 > − µ ( t ) > e ( L ( η ( t )) − /k > e − /k , t > . (5.4)Now, we infer from (3.13) (with η ( t ) = ζ ( t, s ) instead of s ) that ts = η ( t )(1 + tL ( η ( t )))1 − L ( η ( t )) [1 − L ( η ( t )) + k ln (1 − µ ( t ))] s = 1 − µ ( t ) µ ( t ) [1 − L ( η ( t )) + k ln (1 − µ ( t ))] µ ( t ) s =(1 − µ ( t )) (1 − L ( η ( t ))) + k (1 − µ ( t )) ln (1 − µ ( t )) , whence s + (1 − µ ( t )) (1 − L ( η ( t ))) = h (1 − µ ( t )) , t > , (5.5)where h ( z ) := (1 + s ) z + kz ln z , z ∈ ( e − /k , . (5.6)On the one hand, it follows from (5.3), (5.4), and (5.5) thatlim t →∞ h (1 − µ ( t )) = s . (5.7)On the other hand, h is increasing on ( e − /k ,
1) with h ( e − /k ) = se − /k < s < s + 1 = h (1), so that h is a one-to-one function from ( e − /k ,
1) onto ( se − /k , s + 1). Introducing its inverse h − , we deducefrom (5.7) that lim t →∞ µ ( t ) = 1 − h − ( s ) . (5.8) bsence of gelation and self-similarity for a coagulation-fragmentation equation 17 Recalling (5.2) and (5.3), it follows from (5.8) thatlim t →∞ ζ ( t, ts ) t = 11 − h − ( s ) . (5.9)Finally, since L ( t, ts ) = ℓ ( t, ζ ( t, ts )) = L ( ζ ( t, ts )) − k ln (1 − µ ( t ))by (3.12), (3.23), and (5.2), we infer from (5.8), (5.9), and the properties of L thatlim t →∞ L ( t, ts ) = − k ln h − ( s ) . (5.10)We are left with expressing the right hand side of (5.10) with the help of the Lambert W -function.To this end, we note that h − ( s ) solves h − ( s ) (cid:2) (1 + s ) + k ln h − ( s ) (cid:3) = s by (5.6) or, equivalently, W − (cid:0) ln (cid:0) e (1+ s ) /k h − ( s ) (cid:1)(cid:1) = sk e (1+ s ) /k . Therefore, e (1+ s ) /k h − ( s ) = exp n W (cid:16) sk e (1+ s ) /k (cid:17)o , and applying − k ln to both sides of the above identity leads us to (5.1) thanks to (5.10). (cid:3) We finally use [12, Chapter XIII.1, Theorem 2] to express the outcome of Proposition 5.1 in termsof ν and obtain the last statement of Theorem 1.2.6. Second moment estimate
We establish in this section an interesting smoothing property of (1.10), namely that the secondmoment of the solution becomes instantaneously finite for positive times, even it is initially infinite.We also investigate its large time behavior and answer by the positive a conjecture of Vigil & Ziff [37].
Proposition 6.1.
For t > , there holds ∂ s L ( t,
0) = L ( T − ( t ))1 + tL ( T − ( t )) and ∂ s L ( t, ∼ t →∞ − e /k t . (6.1)Note that the positivity of T − ( t ) for t > L ( T − ( t )) isfinite whatever the value t >
0. Recalling that L ( s ) = ( L ( s ) − /s , we realize that L ( T − ( t ))however blows down as t → L ′ (0) = −∞ , that is, f in has an infinite second moment. Proof.
Fix t > θ := T − ( t ). Recalling that L ( t, s ) = ℓ ( t, ζ ( t, s )) by (3.23), it follows from(3.22) that ∂ s L ( t, s ) = ∂ s ℓ ( t, ζ ( t, s )) ∂ s Σ( t, ζ ( t, s )) , s > , Ph. Laurenc¸ot & H. van Roessel while (3.12) and (3.19) give ∂ s ℓ ( t, σ ) = L ′ ( σ ) − ktL ′ ( σ )1 + tL ( σ ) , σ > θ ,∂ s Σ( t, σ ) =1 + tL ′ ( σ ) + k L ′ ( σ ) L ( σ ) [ln (1 + tL ( σ )) − tL ( σ )] , σ > θ . Owing to Lemma 3.4, we may take σ = θ in the previous identities and use (3.17) to obtain ∂ s ℓ ( t, θ ) = L ′ ( θ ) − ktL ′ ( θ )1 + tL ( θ ) = (1 + tL ( θ )) L ′ ( θ ) − ktL ′ ( θ )1 + tL ( θ ) , and ∂ s Σ( t, θ ) =1 + tL ′ ( θ ) + L ′ ( θ ) L ( θ ) ( θL ( θ )) − ktL ( θ ))=1 + tL ′ ( θ ) + θL ′ ( θ ) + 1 − L ( θ ) L ( θ ) − − kt L ′ ( θ ) L ( θ )= (cid:18) t + 1 L ( θ ) (cid:19) L ′ ( θ ) − kt L ′ ( θ ) L ( θ )= (1 + tL ( θ )) L ′ ( θ ) − ktL ′ ( θ ) L ( θ ) . Combining the above formulas for ∂ s L ( t, s ), ∂ s ℓ ( t, θ ), and ∂ s Σ( t, θ ), and recalling that ζ ( t,
0) = θ give the first statement in (6.1) and imply in particular that ∂ s L ( t,
0) is finite.To prove the second statement in (6.1), we use the first one to obtain1 | ∂ s L ( t, | = − t − L ( T − ( t )) = T − ( t ) − t + T ( t ) − L ( T − ( t ))1 − L ( T − ( t )) . Since T − ( t ) ∼ t/ (1 − e − /k ) as t → ∞ by (3.16b), it follows from the properties of L and the aboveidentity that 1 | ∂ s L ( t, | ∼ (cid:18) − e − /k − (cid:19) t as t → ∞ , and the proof of (6.1) is complete. (cid:3) Since ∂ s L ( t,
0) = − Z ∞ yν ( t, dy ) , t > , the second statement (1.15) of Theorem 1.2 readily follows from Proposition 6.1. bsence of gelation and self-similarity for a coagulation-fragmentation equation 19 Acknowledgments
Part of this work was done while PhL enjoyed the hospitality and support of the Departmentof Mathematical and Statistical Sciences of the University of Alberta, Edmonton, Canada and theAfrican Institute for Mathematical Sciences, Muizenberg, South Africa.
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