Smoothness of moments of the solutions of discrete coagulation equations with diffusion
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Abstract
In this paper, we establish smoothness of moments of the solutions of discrete coagulationdiffusion systems. As key assumptions, we suppose that the coagulation coefficients grow at most sublinearly and that the diffusion coefficients converge towards a strictly positive limit (those conditions also imply the existence of global weak solutions and the absence of gelation).
Keywords
Discrete coagulation systems Smoluchowski equations Duality arguments Regularity Smoothness Moments estimatesMathematics Subject Classification
35B45 35B65 82D601 Introduction
In this paper we consider discrete coagulation systems with spatial diffusion. Coagulation models appear in a wide range of applications ranging from chemistry (e.g. the formation of polymers) over physics (aerosols, raindrops, smoke, sprays), astronomy (the formation of galaxies) to biology (haematology, animal grouping), see e.g. the surveys [10, 12, 20] and the references therein.
Following the pioneering works of Smoluchowski (see [26, 27]), we shall denote by \(c_i :=c_i(t,x) \in \mathbb {R}_+\) the concentration of polymers or clusters of mass/size \(i\in \mathbb {N}^*\) at time t and position x. Here, we consider a smooth bounded domain \(\Omega \) of \(\mathbb {R}^N\) in which the clusters are confined via homogeneous Neumann conditions (in applications, we of course have \(N \le 3\)). Moreover, for any positive time T, we denote by \(\Omega _T\) the set \([0,T] \times \Omega \).
Before proceeding further, let us introduce a precise definition of weak solution, following [19].
Definition 1.1

\(c_i\in \mathcal {C}\left( [0,T];L^1(\Omega )\right) \),

\(Q^_i(c)\in L^1(\Omega _T)\),

\(\sup \nolimits _{t\ge 0}\int \nolimits _{\Omega }\rho _1(t,x)dx \le \int \nolimits _{\Omega }\rho _1^{in}(x)dx\),
 \(c_i\) is a mild solution to the ith equation in (1), that iswhere \(Q_i\) is defined by (2), \(A_1\) is the closure in \(L^1(\Omega )\) of the unbounded, linear, selfadjoint operator A of \(L^2(\Omega )\) defined by$$\begin{aligned} c_i(t)=e^{d_iA_1t}c_i^{in} + \int _0^t e^{d_iA_1(ts)}Q_i(c(s))ds, \end{aligned}$$and \(e^{d_iA_1t}\) is the \(\mathcal {C}^0\)semigroup generated by \(d_iA_1\) in \(L^{1}(\Omega )\).$$\begin{aligned} D(A)=\left\{ w\in H^2(\Omega ),\ \nabla w\cdot \nu =0 on \partial \Omega \right\} , \qquad Aw=\Delta w, \end{aligned}$$
The following result, which is a direct application of [19, Theorem 3], states that we can obtain weak solution of (1), (2) from the truncated systems (9), (10). We also refer to [29, 30].
Proposition 1.2
Let \(\Omega \) be a smooth bounded domain of \(\mathbb {R}^N\). Assume that the coagulation coefficients satisfy (3) and that all diffusion coefficients are strictly positive, i.e. \(d_i>0\) \(\forall ~i\in \mathbb {N}^*\). Assume also that the initial concentrations \(c_i^{in}\ge 0\) are such that \(\rho _1^{in}\in L^1(\Omega )\). For every \(n\in \mathbb {N}^*\), let \(c^n=(c_1^n,\ldots ,c_n^n)\) be the solution of the truncated system of size n (9), (10).
Our first proposition states that if the diffusion rates of clusters of different sizes are sufficiently close to each others, the natural uniform \(L^1\)bound (8) can be extended to \(L^p\) (with \(p>1\) depending on the closeness of the diffusion rates). To be more precise about this closeness hypothesis, let us first introduce
Definition 1.3
The existence of such a constant \(\mathcal {K}_{m,q}<\infty \) independent of the time \(T>0\) is explicitly stated in [18] provided that \(\partial \Omega \in \mathcal {C}^{2+\alpha }, \alpha >0\).
Next, we present the
Proposition 1.4
Remark 1.5
Note that this global \(L^2\)bound together with assumptions (3) also ensures that no gelation can occur, so that the conservation law (7) rigorously holds for any weak solution, see [5].
Proposition 1.4 can be improved in the case when the diffusion coefficients \(\left( d_i\right) _{i\in \mathbb {N}^*}\) constitute a sequence converging towards a strictly positive limit. Note that such an assumption is not so far from the assumption that the sequence \(\left( d_i\right) _{i\in \mathbb {N}^*}\) is bounded above and below (by a strictly positive constant), which is used in Proposition 1.4 (or also in [5]), since one expects on physical grounds that the sequence \(\left( d_i\right) _{i\in \mathbb {N}^*}\) is decreasing; that is, that larger clusters diffuse less. Under this assumption and provided that the coagulation coefficients are strictly sublinear (see the precise assumption in Theorem 1.6 below) we can show that \(L^p\) norms of moments \(\rho _k\) are propagated for any \(k \in \mathbb {N}^*\), \(p \in ]1, \infty [\).
Theorem 1.6
Assume that (for some \(k\in \mathbb {N}^*\)) the initial moment \(\rho _k^{in}\) lies in \(L^p(\Omega )\) for all \(p<+\infty \) and that (for all \(i\in \mathbb {N}^*\)) each initial concentration \(c_i^{in} \ge 0\) lies in \(L^{\infty }(\Omega )\).
Then, there exists a global weak nonnegative solution to (1), (2) for which the moment \(\rho _k\) lie in \(L^p(\Omega _T)\) for all \(p<+\infty \) and all finite time \(T>0\).
Remark 1.7
Notice that hypothesis (15) on the coagulation coefficients implies the assumption (3), which in return yields existence of global weak solutions.
Remark 1.8
We point out that Theorem 1.6 could be extended to the case where a finite number of diffusion coefficients \(d_i\) are equal to 0 (see Remark 3.7).
In the case that \(d_i=0\) starting from some \(i=I\) as considered for instance in [31], our approach should allow for a corresponding generalisation of Theorem 1.6 provided that a suitable “closeness” condition on the finitely many nonzero diffusion coefficients is satisfied.
The study of moments for the coagulation equation has been a longstanding strategy to get mass conservation and uniqueness results (see [22] for one of the first work in this direction for the coagulation equation with diffusion, in the continuous case).
The statement of our result is therefore close to that of [25] (our requirement on the diffusion rate is however more stringent), but the proof is completly different, so that the exact conditions required on initial data are also different. Note that the limit case \(a_{i,j} = i + j\) is still open (absence of gelation for this coagulation coefficient is conjectured in general, but is proven only when there is no diffusion, see for instance [4, 28]).
Although in the present work, \(L^p\) estimates for moments are only shown for \(p < \infty \) (whereas \(p= \infty \) can be obtained in [25]), the use of parabolic inequalities for the heat equation enables to recover this case (and also higher order derivatives).
Indeed, the estimates obtained in Theorem 1.6 can be improved if the initial data are assumed to be smooth enough. This leads to our main Theorem, namely:
Theorem 1.9
Let \(\Omega \) be a smooth bounded domain of \(\mathbb {R}^N\). Assume that the coagulation coefficients satisfy (15) and that \(\left( d_i\right) _{i \in \mathbb {N}}\) is a sequence of strictly positive real numbers which converges toward a strictly positive limit.
Assume that the initial data \(c_i^{in} \ge 0\) are of class \(\mathcal {C}^{\infty }(\overline{\Omega })\), compatible with the boundary conditions, and that for all \(k \in \mathbb {N}^*\) the initial moments \(\rho _k^{in}\) are of class \(\mathcal {C}^{\infty }(\overline{\Omega })\).
Then, there exist a unique smooth solution to (1), (2) such that each \((c_i)\) is nonnegative, of class \(\mathcal {C}^{\infty }(\overline{\Omega }_T)\) for any finite time \(T>0\), and such that the moments \(\rho _k\) are also of class \(\mathcal {C}^{\infty }(\overline{\Omega }_T)\), for any \(k\in \mathbb {N}^*\).
Remark 1.10
The \(\mathcal {C}^{\infty }\) regularity down to time 0 requires of course the \(\mathcal {C}^{\infty }\) hypothesis on the initial data. However, it can be seen in the various steps of the proof (see Sect. 4) that propagation of regularity in intermediate Sobolev spaces holds under suitable (less stringent) assumptions on the initial data.
Since each \(c_i\) is solution of a heat equation subject to a r.h.s. that can be controlled once all moments are bounded in \(L^p(\Omega _T)\), \(p<+\infty \), we can in fact show the creation of regularity for strictly positive times. For example, under the assumption that \(\rho _k^{in}\in L^p(\Omega )\) for all \(p<+\infty \) and all \(k\in \mathbb {N}^*\), we can prove that the concentrations \(c_i\) are of class \(C^{\infty }(]0,T] \times \bar{\Omega })\).
Also, as will be made clear in Sect. 4, \(\mathcal {C}^{\infty }\) regularity is not needed to ensure uniqueness. As shown in [15], uniqueness holds as soon as \(\rho _2\in L^{\infty }\), so that starting from initial data leading to an estimate for \(\rho _2\) in a Sobolev space embedded in \(L^{\infty }\), uniqueness can already be obtained.
Finally, we point out that assumption (15) is not far from optimal, since it is known that gelation can occur as soon as \(a_{i,j} = i^{\alpha }j^{\beta }+i^{\beta }j^{\alpha }\) with \(\alpha +\beta >1\) (see [13]) and gelation is not compatible with the conclusion of Theorems 1.6 or 1.9.
Our paper is organized as follows. In Sect. 2, we recall some lemmas existing in the literature and called duality lemmas. We also introduce modified versions of those lemmas, that are later used in Sect. 3 to prove the propagation of moments in \(L^p(\Omega _T)\), \(p<+\infty \) (Propositions 1.4 and 1.6). In Sect. 4, we extend these results to prove \(\mathcal {C}^{\infty }\) regularity for the concentrations and the moments (Theorem 1.9). Finally, a short Appendix is devoted to technical lemmas which are useful to make the proof of some duality lemmas rigorous.
2 Duality estimates
We start by recalling some a priori estimates based on duality arguments from [6]. These estimates are key ingredients of the present work. In this section, functions said to be weak solutions ought to be understood as solutions of the equation obtained by multiplying by a test function and integrating by parts. Remember also that \(\mathcal {K}_{m,q}\) is defined in Definition 1.3.
The first statement recalls [6, Lemma 2.2].
Lemma 2.1
Remark 2.2
The bound on \(\left\ v\right\ _{L^q(\Omega _T)}\) is not explicitly mentioned in Lemma 2.2 of [6], but is a direct consequence of its proof, in particular of the estimates \(\left\ \Delta _x v\right\ _{L^q(\Omega _T)} \le C_1 \left\ f\right\ _{L^q(\Omega _T)}\) and \(\left\ \partial _t v\right\ _{L^q(\Omega _T)} \le C_1 \left\ f\right\ _{L^q(\Omega _T)}\), which are explicitly mentioned there.
Remark 2.3
The fact that the above mentioned function v exists (and is unique) is (in particular for \(q<2\)) not obvious because M is not assumed to be continuous (or at least VMO). In the Appendix (Proposition 4.3), we give a proof of the existence and uniqueness of v for the sake of completeness.
Lemma 2.1 is used to prove the following duality lemma, which is Proposition 1.1 of [6].
Proposition 2.4
In the sequel we will need a generalized version of Proposition 2.4, which is an adaptation of Theorem 3.1 in [11] (where only the case \(p=2\) is treated).
Proposition 2.5
Proof
We finish this section with another variant of the duality lemma, in which \(L^p\) r.h.s. can be treated.
Proposition 2.6
Remark 2.7
We stress the fact that Proposition 2.6 above requires a priori that the function u lies in \(L^p(\Omega _T)\). As a consequence, we shall not be able to directly apply this result to weak solutions of (1), (2), but only to solutions of an approximate (truncated) system (such as (9), (10)), for which we have a priori regularity estimates.
Proof
3 Propagation of moments in \(L^p\) norms
This Section is devoted to the proof of propagation in \(L^p(\Omega _T)\) (\(p<+\infty \)) of moments \(\rho _k\). We begin with Proposition 1.4 and the propagation of the total mass \(\rho _1\), when the closeness hypothesis (13) on the diffusion coefficients is satisfied.
Proof of Proposition 1.4
Remark 3.1
The proof of Theorem 1.6 is a bit more involved but still based on the same idea. The outline of the proof is the following: First, we get \(L^{\infty }(\Omega _T)\) bounds for each concentration \(c_i\) and for any finite time T. Thus, it is sufficient to prove propagation in \(L^p\) spaces for tail moments, in which we only consider concentrations \(c_i\) for i larger than some index I. Because we assumed that the \(d_i\) converge (when \(i \rightarrow \infty \)) towards a strictly positive real number, the closeness hypothesis (13) will always be satisfied for the coefficients \(\left( d_i\right) _{i\ge I}\) when I is large enough. This allows us to use a similar argument as in Proposition 1.4 to prove the propagation in \(L^p(\Omega _T)\) of the mass and then of all higher order moments.
Proof of Theorem 1.6
As for Proposition 1.4, the rigorous way to prove Theorem 1.6 is to get all the needed estimates on the solutions of the truncated problems (9), (10) and then pass to the limit (when \(n \rightarrow \infty \)). However for a clearer exposition of the different arguments, we first derive (sometimes formally) estimates on the whole system (1), (2) and then explain how to pass to the limit in the corresponding estimates on the truncated system. We begin with the following result (which was already noticed in [29]).
Lemma 3.2
Let \(\Omega \) be a smooth bounded domain of \(\mathbb {R}^N\). Assume that the coagulation coefficients satisfy (3) and that \(d_i>0\) for all \(i\in \mathbb {N}^*\). Assume also that each \(c_i^{in}\ge 0\) lies in \(L^{\infty }(\Omega )\). We consider a global weak nonnegative solution of (1), (2) (nonnegative meaning here that \(c_i\ge 0\) for all \(i \in \mathbb {N}^*\)).
Then, the concentration \(c_i\) lies in \(L^{\infty }(\Omega _T)\) for each integer \(i\in \mathbb {N}^*\) and any positive time \(T>0 \).
Proof
The proof of Lemma 3.2 shows sufficient conditions under which each \(c_i\) is bounded on \(\Omega _T\), but explicit bounds computed in this way would grow very fast with i. Thus, there is little hope of obtaining a result on \(\rho _1\) by directly using this method. However, the knowledge that any finite truncation of \(\rho _1\) lies in \(L^{\infty }(\Omega _T)\) enables us to prove another result of propagation of \(L^p\) norms for the mass \(\rho _1\), where the assumption (13) is removed and replaced by the assumption of convergence of the diffusion coefficients \(d_i\) towards a strictly positive limit.
Lemma 3.3
Let \(\Omega \) be a smooth bounded domain of \(\mathbb {R}^N\). Assume that the coagulation coefficients satisfy (15). Assume also that all \(d_i\) are strictly positive, and that \((d_i)\) converges toward a strictly positive limit. Finally, assume that each \(c_i^{in}\ge 0\) lies in \(L^{\infty }(\Omega )\) and that \(\rho _1^{in}\in L^p(\Omega )\) for some \(p\in ]1,+\infty [\). We consider a global weak nonnegative solution of (1), (2) (nonnegative meaning here that \(c_i\ge 0\) for all \(i \in \mathbb {N}^*\)).
Then, \(\rho _1\in L^p(\Omega _T)\), for any finite time \(T>0\).
Proof
Remark 3.4
Note that aside from symmetry, the above proof only requires the estimate \(a_{i,j}\le C\,i\,j\), which is a much weaker restriction on the coagulation coefficients than the “sublinear” assumption (15). However, “strictly superlinear” coagulation is known to produce gelation already in the spatially homogeneous case. Nonetheless, it is known for the homogeneous case that adding sufficiently strong fragmentation in the model can prevent gelation even with “superlinear” coagulation (see for instance [7, 8]). Similar results in presence of diffusion, together with generalisations of some results of this paper to models including fragmentation are discussed in [3].
Continuation of the proof of Theorem 1.6
We shall now prove the propagation of \(L^p\), (\(p<+\infty \)) regularity for moments of higher order. This is done still under the assumption that the diffusion coefficients \(d_i\) converge towards a strictly positive limit.
Remark 3.5
Fatou’s Lemma is enough here to show that \(\rho _k\in L^p(\Omega _T)\), but since for any \(k\in \mathbb {N}^*\) we know that \(\rho _{k+1}^n\) is bounded in \(L^p(\Omega _T)\) uniformlyinn, we could show by interpolation that we do in fact have the convergence of \(\rho _k^n\) to \(\rho _k\) in \(L^p(\Omega _T)\).
Remark 3.6
Note that Theorem 1.6 states that we have propagation of the moment \(\rho _k\) in every \(L^p(\Omega _T)\), \(p<+\infty \), provided that the initial moment \(\rho _k^{in}\) lies in every \(L^p(\Omega )\), \(p<+\infty \). If we only want to get propagation of \(\rho _k\) in \(L^p(\Omega _T)\) for some fixed p, we can relax a bit the hypothesis, but to apply the above proof we still need to assume that initial moment of lower order \(\rho _l^{in}\), \(l<k\), are in some space \(L^q(\Omega )\) with \(q>p\) depending of the magnitude of the coagulation. For instance, if we want to get for some fixed p that \(\rho _2\in L^p(\Omega _T)\) with the method of Theorem 1.6, we need to assume that \(\rho _2^{in}\in L^p(\Omega )\) and \(\rho _1^{in}\in L^q(\Omega )\), where \(q=\frac{2\gamma }{1\gamma }p\).
Remark 3.7
4 Propagation of Sobolev norms for moments
In this Section, we show how the parabolic structure of equation (1) can be used to improve the results of Theorem 1.6 and get higher regularity as stated in Theorem 1.9. We also explain how the obtained regularity in fact implies uniqueness. For some early work on the diffusive coagulationfragmentation equation, using extensively the parabolic structure, we refer the reader to [2].
Proof of Theorem 1.9
We consider a solution provided by Theorem 1.6, for which we already know that we have propagation of moments in \(L^p\) spaces. Remembering (1), we want to use the properties of the heat equation to get additional regularity, and to do so we first need to estimate the coagulation term. This is the content of the following lemma.
Lemma 4.1
Remark 4.2
We remark that analog statements to (31) and (32) could be made in terms of fractional Sobolev spaces, for instance by interpolation arguments.
Proof
Continuation of the proof of Theorem 1.9
Notes
Acknowledgments
Open access funding provided by University of Graz. The research leading to this paper was partially funded by the french “ANR blanche” project Kibord: ANR13BS010004. K.F. was partially supported by NAWI Graz and acknowledges the kind hospitality of the ENS Cachan. The research leading to this paper was also funded by Université Sorbonne Paris Cité, in the framework of the “Investissements d’Avenir”, convention ANR11IDEX0005.
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