Oscillatory dynamics in Smoluchowski's coagulation equation with diagonal kernel

We characterize the long-time behaviour of solutions to Smoluchowski's coagulation equation with a diagonal kernel of homogeneity $\gamma<1$. Due to the property of the diagonal kernel, the value of a solution depends only on a discrete set of points. As a consequence, the long-time behaviour of solutions is in general periodic, oscillating between different rescaled versions of a self-similar solution. Immediate consequences of our result are a characterization of the set of data for which the solution converges to self-similar form and a uniqueness result for self-similar profiles.


Introduction.
Smoluchowski's classical coagulation equation provides a meanfield description of binary coalescence of clusters. If ξ denotes the size of a cluster and F (τ, ξ) the corresponding number density at time τ then the equation is given by where the coagulation kernel K = K(ξ, η) ≥ 0 describes the rate at which clusters of size ξ and η coagulate. This model has been used in various applications, most prominently in aerosol physics or polymerization, but also in astrophysics or population dynamics [1,5,22]. We are in the following interested in kernels K that grow at most sublinearly at infinity. In this case it is well-known for a large class of kernels that, if the initial condition F (0) has finite mass, i.e. finite first moment, then the solution F to (1) conserves the mass for all positive times, that is see the survey papers [12,13,16] and the references therein. Since two clusters merge into one during a coalescence event, the total number of clusters, i.e. the L 1norm of F (τ ), is expected to decay to zero as τ → ∞. This property, together with (2), entails that the mean size s(τ ) at time τ of the distribution of clusters increases without bounds with time but does not reveal much about the dynamics of the coagulation equation (1) and its possible dependence upon the coagulation kernel K. To gain further insight into these matters, the dynamical scaling hypothesis [9,16,23] predicts that, for large times, the solution F (τ ) to (1) behaves in a self-similar way, that is, for some scaling function ϕ to be determined, the prefactor of ϕ being chosen here to comply with the mass conservation (2). Though this issue has been discussed in the physics literature for homogeneous coagulation kernels with homogeneity γ ≤ 1, that is, see [9,16,23] and the references therein, the validity of (3) is still pending, except for the so-called solvable kernels K(ξ, η) = 2 and K(ξ, η) = ξ + η. In these two cases, equation (1) can be solved explicitly by transform methods and a complete characterization of the long-time behaviour of solutions to (1) with solvable kernels can be found in [18]. Let us emphasize that the outcome of the analysis performed in [18] is not only that the dynamical scaling hypothesis (3) is valid for a large class of initial data but also that other self-similar behaviours are possible and involve different time and size scales, as well as different scaling profiles. For all other kernels there are so far only numerical studies available [7,9,11,14] that suggest for a range of kernels convergence to self-similar form in the long-time limit. In subsequent years several results established also rigorously the existence of selfsimilar solutions, both with finite mass and with fat tails [6,8,17,19,20,21], but an analysis of the long-time behaviour of solutions is still elusive. It is worth pointing out that the very first existence results of self-similar solutions for a non-solvable coagulation kernel are obtained for the so-called diagonal kernel where δ is the Dirac mass at zero [17,20]. Observe that the homogeneity of K is equal to γ as the Dirac mass is homogeneous of order −1 (it is the derivative in the sense of distributions of the Heaviside function which is clearly homogeneous of order zero). In this model, only clusters of the same size can coagulate and it is fairly clear that such a model has little physical realization. Nevertheless, it can be viewed as a limit case of a coagulation kernel for which equal size interactions dominate. It has also the interesting feature of reducing the nonlocal nonlinear integral equation (1) to a nonlinear differential equation with delay which is more amenable to analysis, see (6) below, and it is commonly believed that a better understanding of its behaviour paves the way to deal with more general kernels.
In the same vein, we point out that the diagonal kernel is also used in [3,15] to elucidate the onset of gelation, i.e. the infringement of mass conservation, and the results obtained therein were for almost twenty years among the few rigorous ones available on this matter.
We thus focus in this paper on Smoluchowski's coagulation equation (1) with diagonal kernel (5) and we present the first rigorous analysis of the long-time behaviour of solutions for a non-solvable kernel. As already mentioned, it is in some sense orthogonal to the constant one since only clusters with equal sizes interact. When K is given by (5), equation (1) reduces to Obviously a key difference to continuous kernels is the fact that, for the diagonal one, the evolution of the number density F of clusters with mass ξ depends only on smaller values and not on larger ones. As a consequence, the equation obeys a maximum principle, a fact that we will also use in our analysis. Another key feature is that the evolution in ξ depends only on a discrete set of values of the form ξ/2 k and thus the evolution decouples in a certain sense, which leads to a long-time behaviour of solutions that is different from what one expects for continuous kernels. We will explain this in more detail in Section 2.2 below, but point out already here that our results are, to our knowledge, the first ones showing oscillatory behaviour for the solutions with finite mass. Such oscillatory behaviours can also be observed in numerical simulations for kernels which are not diagonal, but are concentrated near the diagonal [10]. These results indicate that the self-similar behaviour is not the only possibility for the coagulation equation, although this has been often assumed and conjectured.
Then G satisfies and Well-posedness of (7) for initial data in L ∞ (R) will be shown later for a reformulated problem in Lemma 3.2. We expect to have as special solutions to (6) self-similar solutions of the form which means for (7) rescaled traveling wave solutions of the form In order to study the existence of such special solutions and the large time behaviour of solutions to (7) we introduce the new variables α := (1−γ) ln 2 > 0 , t := ln τ α , x := η − ln τ α , e −αt h(t, x) := αG(τ, η) , (9) and we deduce from (7) and (8) that h solves A self-similar solution to (6) corresponds now to a stationary solutionh of (10), that is, to a solution of It is proved in [17] by a shooting method that for any a > 0 there exists a nonnegative solutionh a ∈ L 1 (R; e αx dx) to (12) that is bounded, decreasing and satisfies where σ is the only positive root of the equation The constant a is just a normalization and can be chosen to adjust the mass R e αxh a (x) dx. Equivalently, since equation (12) is invariant under a shift, we can take an arbitrary a > 0 and then shift the correspondingh a to obtain any positive mass. From now on,h ∈ L 1 (R; e αx dx) denotes a fixed solution to (12) with Notice also that it follows directly from (12) and the local integrability ofh that h ∈ C 1 (R). Our goal in this paper is to study the behaviour of solutions to (10) as t → ∞ and to figure out what is the role played by the family (h a ) a>0 , if any. As a consequence of our results we will actually show that stationary solutions are unique up to rescaling.

Main results.
A key property of the diagonal kernel is that the evolution of h(t, x) depends only on the discrete set of values x − k, where k ranges in the set N 0 of nonnegative integers. Correspondingly, one of the key points of the argument used in this paper is a decomposition of the plane (t, x) ∈ R 2 in a family of lines whose evolution under (10) is mutually decoupled.
More specifically, given θ ∈ [0, 1), we define the following family of lines: Obviously for given Hence, the following function Θ is well-defined.
Here, we use the notation {y} := y − y where · denotes the usual right-continuous floor function.
Notice that the function Θ is 1-periodic in both variables, that is We also define the function Notice that, in particular, ψ(0, θ) = 1 − θ for θ ∈ (0, 1) and that t → ψ(t, θ) is right-continuous and jumps from 1 to 0 at times n + θ, n ∈ N, Introducing we infer from (12) and (14) that Consequently, ν(θ) = const. for θ ∈ [0, 1) and, since (15) can be rewritten as the normalization (19) Then, in order to obtain fromh a stationary solution to (10) with mass M > 0, we need to shifth and see that the shift λ is determined by the following relation: As pointed out above, the evolution of (10) decouples into an evolution for each 'fibre' S θ , θ ∈ [0, 1). We shall see that, given a nonnegative initial condition h 0 ∈ L ∞ (R), the long-time behaviour of the corresponding solution to (10) is determined by the mass of the initial condition in each fibre, given by More precisely, we will see that a solution of (10) is for large times approximated in each fibre θ ∈ [0, 1) by the stationary solution with the same mass, that is, according to (21), the shifth(· − λ(θ)) ofh such that We can now formulate our main results.
Theorem 2.2. Consider the solution h of (10) with a nonnegative initial condition Theorem 2.2 provides a detailed description of the asymptotic behaviour of solutions to (10) as t → ∞ and in particular implies that the long-time behaviour is in general periodic. We note that periodic long-time behaviour is also observed in a growth-fragmentation model where the fragmentation kernel is diagonal [2]. Theorem 2.2 also characterizes the class of initial data with finite mass that yield a traveling wave solution (resp. self-similar solution in the original variables) as t → ∞. Corollary 1 applies in particular to functions h 0 given by h 0 (x) := H k e −α(x−k) for x ∈ [k, k + 1) and k ∈ Z, where (H k ) k∈Z is a sequence of positive real numbers which is bounded and such that H k e αk converges. For instance, H k = e −βk+ , β > α, or H k = (k + ) −q e −αk+ , q > 1, recalling that k + := max{k, 0}. Let us nevertheless emphasize here that, according to Theorem 2.2, solutions emanating from arbitrarily small perturbations of such initial data may have a drastically different behaviour in the long term, as soon as the corresponding function m 0 in (22) is not constant.
Another consequence of Theorem 2.2 is the fact that stationary solutions of (10) are unique up to rescaling. Proposition 1. Letĥ andh be two nonnegative solutions to (12) The remainder of the paper is organized as follows. Section 3, which is the core of our paper, contains the proof of Theorem 2.2 and consists of several lemmas.
The key idea is that we can construct a Lyapunov functional L θ for the evolution in each fibre that resembles a contractivity property of solutions to scalar conservation laws and nonlinear diffusion equations [4,24], see Lemma 3.4. We also identify its limit for large times in Lemma 3.6. Another key auxiliary result is a tightness property that we prove in Lemma 3.3. This in turn allows us to obtain a lower bound for the dissipation functional in Lemma 3.5. All these results are proved pointwise in the sense that we prove them for each fibre. Corollary 1 is then a direct consequence of Theorem 2.2. Finally, in order to apply Theorem 2.2 to deduce the uniqueness result stated in Proposition 1 we need an a priori L ∞ bound for arbitrary nonnegative solutions to (12). The corresponding result provided in Lemma 4.1 and the conclusion are the contents of Section 4. (22) belongs to L 1 (0, 1). In particular, m 0 (θ) is finite for almost every θ ∈ [0, 1). Let h be the solution to (10) with initial condition h(0) = h 0 at time t = 0.
for some C 0 > 0. Then there exists a unique function ϕ θ ∈ C 0 [0, ∞); Y + with ϕ θ k ∈ C 1 J θ for all k ∈ Z which solves (26) and (27) with initial condition ϕ θ (0) and satisfies (29) as well as the following property: We remark that (31) is based on a comparison argument and thus the bound depends on ϕ θ (0) l∞ , but not on the mass m 0 .
Proof. Throughout the proof we omit the dependence on θ in the notation. It turns out to be convenient to go over to the unknown function Then we are looking for a solution of and k∈Z e αk ρ k (t) = m 0 (θ) for all t ≥ 0 .
To proceed further, we set which is a right-continuous function of time, and notice that (24) and (27) give Our goal is to show that L θ (t) → 0 as t → ∞. This will follow from the fact that L θ is almost a Lyapunov functional and that we can provide a lower bound on the dissipation functional. Towards that aim we first prove a tightness result.
and |k|≥N e αkφθ Proof. Within this proof we again omit the dependence on θ in the notation. We first notice that the uniform bound (31) and the boundedness ofh imply that there exists N 1 ∈ N such that In order to control the mass at large positive k we construct a supersolution for the quantiles. More precisely, for t ≥ 0, we define with some sufficiently large 0 ∈ N that will be determined later. Owing to (26) we note that Q k solves andQ k solves the same equation for k ∈ Z. Furthermore, by (24) and (27), we have Q k ((n + θ) − ) = e α Q k−1 (n + θ) andQ k ((n + θ) − ) = e αQ k−1 (n + θ) for n ∈ N and k ∈ Z. We also observe that, by (29), lim k→−∞ Q k (t) = e α(ψ(t)−1) m 0 and lim k→−∞Q k (t) = e α( 0+ψ(t)) m 0 + εe αψ(t) , t ≥ 0 .
For the difference W k := Q k −Q k we obtain we infer from (24) and (31) that In addition, for K ∈ N large enough. We then deduce from (41) and (42) that and thus, after integration, for t ∈ [(n − 1 + θ) + , n + θ) and n ∈ N. We now choose 0 = 0 (M, m 0 ) so large such thatQ Also, by (38), we haveQ With this choice of 0 , we conclude that W k (0) ≤ 0 for all k ∈ Z, hence W k (t) ≤ 0 for t ∈ [0, θ) and k ∈ Z ∩ (−∞, K] by (43). Since K ∈ N is arbitrary we have thus proved that Q k (t) ≤Q k (t) for t ∈ [0, θ) and k ∈ Z. In particular, , which allows us to iterate the above argument and end up with Now, according to (24),Q k is 1-periodic and, for t ∈ [0, 1), we infer from (14) that for k ≥ N 2 (θ) sufficiently large. Choosing N = max{N 1 , N 2 } finishes the proof.
Proof. This is a simple explicit computation.
Assume for contradiction that there exists a sequence (p m ) m≥1 in T ε,N such that where p m = (p m k ) k∈Z , W m k := p m k −φ k , and Owing to (26) and (31), (p m k ) m≥1 is bounded in C 1 ([θ, θ + 1)) for all k ∈ Z and we can extract a subsequence, again denoted by (p m ) m≥1 such that (p m k ) m≥1 converges uniformly in [θ, θ + 1) to a function p k for all k ∈ Z. Setting p := (p k ) k∈Z , we easily see that p is a solution to (26) and satisfies the uniform bound (31) and the tightness bound (39) in [θ, θ + 1). In addition, due to the tightness property (39), we deduce from the mass equation (29) and the lower bound (44) which are valid for p m that k∈Z e αk p k (t) − e α(ψ(t)−1) m 0 ≤ 4e α ε and |k|≤N e αk W k (t) ≤ 4e α ε , (46) with W k := p k −φ k , as well as for t ∈ [θ, θ + 1). Now, since each term in the sum D m is non-negative, there holds It remains to take the limit in (48). Fix k ∈ Z ∩ [−N, N ]. Introducing b k := e αk p k +φ k W k sign + (W k ) − sign + (W k+1 ) , k ∈ Z , and the sets B k := {t ∈ (θ, θ + 1) : W k (t) > 0 and W k+1 (t) = 0} , we have (θ, θ + 1) = B k ∪ C k ∪ G k and we notice that the convergence of (p m k ) m≥1 and the definition of W m k imply that lim We now claim that |B k | = 0. To prove this we use the equation satisfied by W k+1 which reads, due to (26), for t ∈ (θ, θ + 1). Let t 0 ∈ B k . By (51) Consequently, there is δ > 0 such that B k ∩ [t 0 − δ, t 0 + δ] = {t 0 } and B k contains only isolated points. This implies that |B k | = 0 and therefore (50) reduces to This in turn implies that We will now show that the functions p k and the corresponding W k constructed above cannot exist. Indeed, assume first that there is t * ∈ [θ, θ + 1) such that W −N (t * ) > 0. Due to (46) and (47), W k (t * ) cannot be positive for all k ∈ Z ∩ [−N, N ] and we define Then W k * (t * ) > 0 and W k * +1 (t * ) ≤ 0. Since by (51), we realize that for some δ > 0. Reducing δ if necessary, we may also assume that W k * (t) ≥ W k * (t * )/2 for t ∈ [t * − δ, t * ). This implies that for t ∈ [t * − δ, t * ) and contradicts (53). Therefore W −N (t) ≤ 0 for all t ∈ [θ, θ + 1). We fix any value t * ∈ [θ, θ + 1) and define . Then k 1 > −N and (47) guarantees that k 1 ≤ N . Since W k1−1 (t * ) < 0 and W k1 (t * ) ≥ 0, we use once more (51) to obtain We may then find δ > 0 sufficiently small such that This implies that for t ∈ [t * − δ, t * ) and contradicts again (53). This concludes the proof. Proof. We first note that, for t ∈ J θ , Lemma 3.4 ensures that d dt L θ (t)e −αt = −e αt D θ (t) .

4.
A priori estimates for stationary solutions.
A by-product of the proof of Lemma 4.1 is the explicit bound h L ∞ (R) ≤ 8e 2α , but it is likely to be far from optimal. Proof.
Step 1. We first claim that the functionh satisfies To prove (59), we first multiply (12) by e αx and integrate to find for some J ∈ R. In particular, e αxh (x) ≥ J and the integrability of x → e αxh (x) implies that J ≤ 0. It next follows from the integrability of x → e αxh (x) that there is a sequence (x n ) n≥1 in (1, ∞) such that lim n→∞ x n = ∞ andh(x n − 1) ≤ e α(xn−1)h (x n − 1) ≤ 1 2 , n ≥ 1 .
We are going to show that the differential equation (12) forh guarantees thath does not exceed one in (x n − 1, x n ). Indeed, by (12), ∂ xh ≤h 2 and integrating this equation gives, thanks to (61), (z)e αz dz .
Since the right-hand side of the above inequality converges to zero as n → ∞, we conclude that J = 0 which proves (59).
Step 2. To complete the proof we argue by contradiction and assume that there is M ≥ 8e 2α and x 0 ∈ R such thath According to (59) we have either We have thus proved thath Arguing by induction, we actually conclude that Furthermore, arguing as in the proof of (66), we infer from (59) and (67) that where either I k := (x 0 + (k − 1)/2, x 0 + k/2) or I k := (x 0 + k/2, x 0 + (k + 1)/2). Define L := k ∈ N : I k = x 0 + k − 1 2 , x 0 + k 2 and R := k ∈ N : Then L ∪ R = N and Since L or R is infinite, the right-hand side of the above inequality cannot be bounded, which gives a contradiction and hence proves the statement of the lemma.