MODELING DEGRADATING DISPERSIONS IN A THREE-DIMENSIONAL FINITE CONTAINER UNDER GENERAL BOUNDARY...

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

SIAM J. MATH. ANAL. c© 2013 Society for Industrial and Applied MathematicsVol. 45, No. 4, pp. 2332–2353

MODELING DEGRADATING DISPERSIONS IN ATHREE-DIMENSIONAL FINITE CONTAINER UNDER GENERAL

BOUNDARY CONDITIONS∗

ROBERTO GIANNI† AND FABIO ROSSO‡

Abstract. This paper deals with a mathematical model widely used to analyze the evolution ofa dispersion of bubbles in a liquid. The model takes into account both diffusion and buoyancy in afinite three-dimensional container. Rather general Dirichlet as well as Neumann boundary conditionsare allowed. We prove the well-posedness of the initial-boundary value problem. The unique solutionis classical and global in time.

Key words. integral-differential equations, Dirichlet and Neumann boundary conditions

AMS subject classifications. 45K05, 35R09, 35k99

DOI. 10.1137/120894087

1. Introduction. The classical mathematical model for a quietly degradatingdispersion is an integral-differential equation continuously parametrized by a positivescalar v which represents the drop volume of the guest continuum. These systemsare generally unstable because of the density difference. The other main mechanismdriving the instability is coalescence: two colliding drops may form a single one.Diffusion also plays a role and therefore should be taken into account. The relativeimportance of any single mechanism with respect to each other depends obviouslyupon the real nature of the constituents.

A continuously increasing mathematically oriented literature has been devoted toselected aspects of the general problem during the last 60 years. The problem hasseveral industrial applications but we do not focus directly on them. Our interest inthe present paper is to investigate the model from a purely mathematical point of viewunder rather general boundary conditions. The reason for our choice is mainly thatthe whole problem and the modeling approaches are still in progress and contributionsin this area are generally welcome, since important issues remain open problems. Forexample, drop breakage under shear stress is still under investigation, and similarlythere are several different descriptions of the coalescence phenomenon. In some casesmodels do not guarantee mass conservations; see [17]. Moreover, as far as we know,there are practically no mathematically oriented papers which model and analyzethe evolution of the system once the phase separation has begun, which is a furtherinteresting open problem.

Since the evolutions of these systems are far from being trivial, mathematicianshave been analyzing the well-posedness of the proposed models for half a century (see,e.g., [16]) and this kind of literature has grown significantly.

∗Received by the editors October 8, 2012; accepted for publication (in revised form) May 15,2013; published electronically August 6, 2013.

http://www.siam.org/journals/sima/45-4/89408.html†Dipartimento di Metodi e Modelli Matematici, Facolta di Ingegneria, Universita di Roma La

Sapienza, Via Antonio Scarpa 16 (palazzina B), 00161 Roma ([email protected]).‡Dipartimento di Matematica e Informatica Ulisse Dini, Universita degli Studi di Firenze, Viale

Morgagni 67/a, 50134 Firenze, Italia ([email protected]).

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MODELING DEGRADATING DISPERSIONS 2333

The aim of this paper is to investigate the basic evolution model with coalescence,buoyancy, and diffusion (but no breakage), considering both the case of Dirichletboundary conditions and that of Neumann boundary conditions.

To be more precise, we assume that the emulsion fills completely a finite containerΩ ⊂ �3 (even if our technique applies to �N for any positive integer N).

We distinguish the concentration S of the guest continuum from the volumetricdistribution of drops n, which depends, besides the volume v of the drops and timet, only on the spatial coordinate x. Obviously, drops cannot exceed a given (finite)maximum volume vmax. Thus S = S(x, t), n = n(x, v, t), and

(1.1) S(x, t) =

∫ vmax

0

vn(x, v, t)dv.

One expects that 0 ≤ S ≤ 1 everywhere in the physical domain and at anytime. We were not able to prove it in the fully general case in which the diffusioncoefficient D and the buoyancy velocity V depend on the droplet volume v, but it canbe easily achieved if D = D(S,x, t) and V = V(S,x, t) with the additional conditiondivx V = 0. (See the concluding remarks section at the end of the present paper.)Moreover, since S(x, t)dx is the amount of guest continuum in the elementary volumecentered in x at time t, then

(1.2) Φ(t) =

∫Ω

S(x, t)dx =

∫Ω

∫ vmax

0

vn(x, v, t)dvdx

provides the total amount of dispersed continuum. Clearly Φ(t) = constant if andonly if the container is isolated.

We notice explicitly that, when we consider an isolated container, volume con-servation needs to be a direct consequence the system of equations aimed to modelthe evolution of liquid emulsions. This aspect of the problem has been deeply andexhaustively investigated in several papers (see, e.g., [2, 3, 4, 5, 7, 12]) regardless ofwhether diffusion is taken into account.

In our approach we allow very general Dirichlet as well as Neumann boundaryconditions, so that Φ(t) is not to be expected, in general, to remain constant. Our aimis to focus on the existence and uniqueness, globally in time, of a classical solutionwhen phase separation has not yet produced a markedly distinguishable separationboundary. Our approach is not completely general as far as the dependence of thediffusion coefficient on S is concerned. Nevertheless we think that the present paperis a reasonable contribution to actual discussions and investigations on this subject.From this point of view, our analysis should be compared with the papers of [1, 2,3, 15]. However, a sharp comparison may turn out to be difficult in some cases.For example, in [1] the region of motion identifies with the whole �N so that noboundary conditions exist. Moreover, the solution is global in time only for n = 1or if the diffusivity coefficient does not depend on the droplet volume or, finally, ifcoagulation does not take place. A slightly better comparison can be made with paper[2], although the authors use rather different techniques. In this case the authorsconsider a finite container. However, the sole boundary condition allowed is thatof zero flux. Moreover, buoyancy is not considered and, although a fully generalcoalescence-breakage-scattering operator is taken into account, throughout that paper,globally-in-time existence is achieved only if there is no fragmentation and if a suitableLp-norm of the initial data is sufficiently small.

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2334 ROBERTO GIANNI AND FABIO ROSSO

Paper [15] also deals with the problem of global existence. Also in this case thetechniques used are rather different from ours and a limited comparison can be madeonly as far as the main result is concerned. Indeed, besides the nature of hypotheses(zero boundary flux and a strong assumption on the structure of the coalescencekernel) which are quite different from ours, the authors show the globally-in-timeexistence of weak solutions, while we can achieve the same goal for strong solutions(even though under stronger differentiability conditions). Moreover, our assumptionson the coalescence kernel (clearly stated by hypotheses A.1, A.2, and D.2 in the paper)consist exclusively of boundedness or regularity requirements. To be more precise,what we really need is the boundedness from above of all the relevant coefficients(coalescence kernel, diffusion coefficient, and the magnitude of the buoyancy velocity)and, just for the diffusion coefficient, also bounded from below by a positive constant.This last assumption is indeed unavoidable in order to guarantee the strict parabolicityof the main evolution operator. On the other hand, assumptions on the diffusioncoefficient are indeed quite common in the literature on this subject (see, e.g., [1, 2,3, 15]).

Moreover, the novelty of our approach is the a priori L∞-estimate of the solution(Proposition 3.3 in the present paper). This is the key point to get the existence theo-rem. This estimate relies in a fundamental way upon the structure of the coalescenceoperator which allows us to treat the integral operator as if it were linear even thoughit is not (see inequality (3.3)).

The fact that we confine ourselves to coalescence as a leading mechanism fordegradation may appear to be a limitation. We decided to drop the breakage effectmainly for physical reasons since, when density difference is the only driving mech-anism of degradation, collisions (due to different buoyancy velocities or to Browniandiffusion) resulting in fragmentation are hardly observed. However, the adding ofbreakage (which is a linear operator with respect to n) is not a source of mathemati-cal complications and could be done without too much effort.

2. Mathematical setup: The case of Dirichlet boundary conditions. Wetreat first the case of Dirichlet boundary conditions.

We introduce the oil volume fraction S and the volumetric distribution of oildrops n per unit volume. The whole problem will be treated, more generally, in anN -dimensional setting, that is, we allow n to depend, besides on the volume v ofthe drops and on time t, on the spatial coordinate x ∈ Ω ⊂ �N (clearly N = 3 isthe only physical interesting case), being Ω a finite container with smooth boundary(∂Ω ∈ C2), completely filled by the emulsion. Thus S = S(x, t) and n = n(x, v, t).

We also use the following set notation:

Υv := {(x, w) | x ∈ Ω, w ∈ [0, v]} ,Ω t := {(x, τ) | x ∈ Ω, τ ∈ (0, t)} ,Θt,v := {(x, w, τ) | x ∈ Ω, τ ∈ (0, t), w ∈ [0, v]} .

We now write explicitly the mathematical model for the evolution of the popu-lation of droplets which includes both buoyancy as well as diffusion. The model hasbeen proposed many times in different research papers: some do not take into accounteffects like diffusion and buoyancy but consider other physical important effects likefragmentation and scattering (see, e.g., [1, 7, 8, 9, 13, 16]). Other papers includediffusion (see, e.g., [2, 3, 15, 18]). However, the list of relevant references is ratherlong: the citations above are just a sample for an argument which has many intriguingaspects from both the mathematical and the physical point of view.

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The system of equations for n considered in the present paper is the following:

(2.1) ∂tn(x, v, t) +∇ · j(x, v, t) = L[n]

with boundary conditions

n(x, v, t) = φ(x, v, t), x ∈ ∂Ω, t ∈ [0, T ],(2.2)

n(x, v, 0) = nin(x, v), x ∈ Ω,(2.3)

where

(2.4) j(x, v, t) := V (S(x, t), v)n(x, v, t)−D(S(x, t), v)∇n(x, v, t)

(2.5)

L(n; v) =1

2

∫ v

0

n(x, v − w, t)n(x, w, t)Q(S(x, t); v − w,w)dw

−n(x, v, t)

∫ vmax−v

0

n(x, w, t)Q(S(x, t); v, w)dw.

In the very physical problem V represents the upward particle velocity due tobuoyancy forces, Q is the (symmetric) aggregation kernel, and D is obviously a diffu-sion coefficient. Finally, the coalescence operator L(n; v) describes the gain of droplets(the first term) and the loss (the second term) at level v.

As we stated in the introduction, the purpose of this paper is to prove existenceand uniqueness of a classical solution of problem (2.1)–(2.3). To this aim we needto prove some a priori estimates, the first of which will be proved in the followingsection.

3. A priori estimates for Dirichlet boundary conditions. We devote thissection to produce some a priori estimates for the “classical solution” of the system ofequations (2.1)–(2.3). Existence and uniqueness of such a classical solution (at leastin a sufficiently small time interval [0, T1] with T1 < T ) will be proved in section 4 bymeans of a contraction mapping technique. In turn the a priori estimates presentedin this section will allow us to iterate the local existence theorem of section 4, thusproving existence and uniqueness in the whole time interval [0, T ].

We first prove an L∞-estimate.To this purpose we need the following assumptions.

(A.1) Q ≡ 0 if v + w > vmax.(A.2) 0 ≤ Q ≤ Qmax < ∞.(A.3) |V| ≤ K < ∞.(A.4) 0 < K−1 ≤ D ≤ K.(A.5) φ ≥ 0 and φ ∈ L∞([0, vmax];C

0(∂Ω× [0, T ])).(A.6) n(x, v, 0) = nin(x, v) ≥ 0, nin ∈ L∞([0, vmax];C

0(Ω)), x ∈ Ω, where nin

does not vanish identically in Υvmax .(A.7) The zeroth-order compatibility condition is satisfied for all v ∈ [0, vmax].

In what follows we refer to the whole set of the above hypotheses simply as theA hypotheses.

Proposition 3.1. Under assumptions A, any classical solution of (2.1) is non-negative in ΘT,vmax .

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Proof. We consider a modified problem in which L is replaced by

L+(n) =1

2

∫ v

0

n+(x, v − w, t)n+(x, w, t)Q(S(x, t); v − w,w)dw

−n(x, v, t)

∫ vmax−v

0

|n|(x, w, t)Q(S(x, t); v, w)dw.

We will prove that any classical solution of the modified problem is nonnegative. Onthe other hand, if n is nonnegative, the modified and the original problem identify;hence the result follows provided that a uniqueness theorem holds for the originalproblem (what will be shown in section 4). To prove the claim, multiply (2.1) (withL+ replacing L) by n− and integrate over Ω t to get

1

2

∫Ω t

∂t(n−)2ds+

∫Ω t

∇ · (n−j)ds−∫Ω t

j · ∇(n−)ds =

∫Ω t

n−L+(n)ds,

where ds := dxdt. Since Q ≥ 0 it is obvious that n−L+(n) ≤ 0. Moreover, beingn(x, v, 0) = nin(x, v) ≥ 0 we obtain the inequality

1

2

∫Ω

(n−(x, t))2dx−∫ t

0

∫∂Ω

[(V n−D∇n) · νn−] dt

+

∫Ω t

D(∇n−)2ds−∫Ω t

V n− · ∇n−ds ≤ 0,

ν being the outward unit normal to ∂Ω, which implies, taking into account φ ≥ 0,∫Ω

(n−(x, t))2dx+

∫Ω t

D(∇n−)2ds ≤∫Ω t

V 2

D(n−)2ds ∀t > 0.

Then a standard application of Gronwall’s lemma implies that n− ≡ 0 (for the mod-ified problem), which concludes the proof.

Proposition 3.2. Under assumptions A, for all T > 0 there exist M1(T, v) andv(T ) ≤ vmax such that |n(x, t, v)| ≤ M1(T, v) in ΘT,v.

Proof. As a preliminary step let us observe that for a sufficiently small T1 < Twe have

(3.1) maxΘT1,vmax

|n(x, t, v)| ≤ 2I,

where I := maxΥvmax|nin(x, v)|, where obviously T1 is not controlled at all (in the

sense it is not bounded from below in terms of known quantities). Indeed (3.1)follows from the fact that n is a classical solution; in particular we assume n ∈L∞([0, vmax];C

2,1(ΩT )).Then we have that 0 ≤ n ≤ n, where n solves

∂tn+∇ · ([V (n)]n− [D(n)]∇n)

=1

2

∫ v

0

Q(S, v − w,w)n(v − w)n(w)dw := [L(n; v)](3.2)

with the boundary conditions (2.2)–(2.3). [f(n)] is used to recall that function f ,whatever it is, is meant to be evaluated on n, not on n. Notice that [L] contains only

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the “production” term, and because of the positivity of Q and n, we have [L] ≥ [L].In this regard, (3.2) together with its boundary conditions has to be read as a linearequation in n, the terms V , D, and the operator L of (3.2) being viewed as calculatedon n which, at this step, is supposed to be known. Function n, as a solution of(3.2) with the boundary conditions (2.2)–(2.3), is clearly a supersolution of the realproblem.

Let us now define M(T, v) := maxΘT,v|n(x, t, v)|. Obviously[

L(n; v)]≤ QmaxM

2v

if v ∈ [0, v].We now use Theorem 7.1 of [14, p. 181], thus obtaining an L∞ estimate of n.

It is important to stress that this estimate depends on QmaxM2v but, since (3.2) is

regarded as linear (D, V being bounded quantities), we get that the estimate dependslinearly on the quantity QmaxM

2v (which is an upper bound for L(n; v)), i.e.,

0 ≤ n ≤ n ≤ C0 + C1QmaxM2v ∀v ∈ [0, v],

where C0, C1 depend only on the boundary data and on K from assumption A. Thisin turn implies

(3.3) M ≤ C0 + C1QmaxM2v.

Without loss of generality we can assume C0 > 2I (recall that I := maxΥvmaxnin).

We claim that 0 ≤ n ≤ 2C0 in ΘT,v, provided v is chosen sufficiently small. Indeed,

because of the continuity of n, the set Tv := {T | n(x, t, v) ≤ 2C0, (x, t) ∈ ΩT },with v < v, is closed. It is also not empty since any T < T1 belongs to this set.Finally it is open: indeed, from (3.3), we have that

(3.4) M ≤ C0 + C14C20Qmaxv ≤ 3

2C0 in Tv,

provided that v ≤ 1/(8QmaxC20C1) = b. Hence, using again the continuity of n, we

get that if T ∈ Tv, then T + δ ∈ Tv for a sufficiently small positive δ. Since Tv isboth a closed and an open nonempty subset of [0, T ], necessarily Tv = [0, T ], and,since v can be arbitrarily chosen, provided that v < v, the theorem is proved if wetake M1 = 2C0.

Proposition 3.3. Under assumptions A, for all T > 0, (x, t, v) ∈ ΘT,vmax ,

(3.5) |n(x, t, v)| ≤ M(T ) ∀v ∈ [0, vmax].

Proof. We will use a kind of inductive procedure onm ∈ �, working for (x, t) ∈ ΩT

and v ∈ [0,mv∗], with v∗ to be chosen later. We have just proved that |n| ≤ M1 =2C0 in ΩT when v ∈ [0, v∗] (with v∗ ≤ b = 1/(8QmaxC

20C1)). We recall that with

Mm we denote a generic constant depending only on the boundary data and on thecoefficients via the constant K. The claim of the theorem is true for m = 1 becauseof Proposition 3.2. Let us assume that the same claim is true at index level m (i.e.,|n| ≤ Mm in ΩT , v ∈ [0,mv∗]) and let us prove the existence of a bound for n whenv ∈ [0, (m+1)v∗] (provided that (m+1)v∗ ≤ vmax). Obviously we have only to provethe assertion for v ∈ [mv∗, (m + 1)v∗]. To this purpose let us observe that the first

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integral appearing in the coalescence operator (omitting the inessential dependencies)can be split this way:

(3.6)

1

2

∫ v∗

0

Q(w, v − w)n(w)n(v − w)dw +1

2

∫ v−v∗

v∗Q(w, v − w)n(w)n(v − w)dw

+1

2

∫ v

v−v∗Q(w, v − w)n(w)n(v − w)dw.

It is now quite easy to apply the induction hypothesis to get

(3.7) L(n; v) ≤ 1

2QmaxvmaxM

2m +Qmaxv

∗Mm+1M1 := Bm

if v ∈ [mv∗, (m + 1)v∗]. Indeed, while the term M2m in the right-hand side of (3.7)

is quite obvious, the second term comes from the first and the last integral of (3.6);in particular, the factor M1 comes from the term n(w) in the first integral of (3.6)and the term n(v −w) in the last integral of (3.6) too. Proceeding as in the previousproposition we get an estimate of n in ΩT with v ∈ [mv∗, (m+ 1)v∗] which dependslinearly on Bm (in the sense specified in Proposition 3.2); indeed we get a bound fromabove, but this is sufficient since n was already proved to be positive. Hence we get

maxXm,T

|n| ≤ D0 +D1Qmaxv∗M1Mm+1

where Xm,T = ΩT × [mv∗, (m + 1)v∗]. Here D0 depends only on known quantitiesand on Mm (i.e., it does not depend on Mm+1), while D1 depends only on knownquantities. This implies

Mm+1 := maxΩT ×[0,(m+1)v∗]

|n| ≤ Mm +D0 +D1Qmaxv∗M1Mm+1.

If v∗ is chosen such that

v∗D1QmaxM1 ≤ 1

2

we get

(3.8) Mm+1 ≤ 2(D0 +Mm).

Inequality (3.8) is a recursive relation implying that a global L∞ estimate holds inΩT × [0, vmax] since the interval [0, vmax] can be covered with a finite number (i.e.,[vmax/v

∗]+ 1) of steps of width v∗. Hence there exists a finite bound M(T ) such that(3.5) holds true.

We now prove that if nin and φ are initially zero in a small right neighborhoodof v = 0, then n remains so as long as it exists.

Proposition 3.4. Let n(x, v, t) ≡ 0 for all (x, t) ∈ ∂pΩT (the parabolic boundary,see Appendix B) and for all v ∈ [0, vmin]. Then n(x, v, t) ≡ 0 for all (x, v, t) ∈ΘT,vmin .

Proof. Let us multiply (2.1) by n and integrate w.r.t. x and t over Ωt. By theboundary conditions it follows, for v ∈ [0, vmin], that

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1

2

∫Ω

n2(x, t, v)dx(3.9)

≤ Qmax

2

∫Ωt

[n(x, t, v)

(∫ v

0

n2(x, v − w, t)dw

)1/2

×(∫ v

0

n2(x, w, t)dw

)1/2]ds+ C

∫Ωt

n2(x, t, v)ds

≤ C1

∫Ωt

n2(x, v, t)ds+Qmax

2

∫Ωt

(∫ vmin

0

n2(x, t, w)dw

)2

ds

≤ C1

∫Ωt

n2(x, v, t)ds+Qmax

2M2vmin

∫Ωt

(∫ vmin

0

n2(x, w, t)dw

)ds,

where M := maxΘT,vmin|n|. Integrating the last inequality w.r.t. v in [0, vmin] and

setting y(τ) :=∫Ω

∫ vmin

0 n2(x, v, τ)dvdx we get

(3.10) 0 ≤ y(t) ≤ (2C1 +QmaxM2v2min)

∫ t

0

y(τ)dτ,

which implies y ≡ 0, i.e., n ≡ 0 in ΘT,vmin .

Remark 1. If the assumptions on the boundary data guarantee that Proposition3.4 holds, then Proposition 3.3 can be proved more easily and in a quicker way. Indeedthe bound M1 appearing in Proposition 3.2 (and in the proof of Proposition 3.3) isequal to zero provided that b is chosen equal to vmin; hence (3.7) can be modified,obtaining

L ≤ 1

2QmaxvmaxM

2m

for v ∈ [mvmin, (m + 1)vmin]. Having thus estimated L in terms of Mm, proceedingas in Propositions 3.2 and 3.3, we get that

(3.11) Mm+1 ≤ F (Mm),

where F is a positive increasing function of its argument. At this point the globalestimate required by Proposition 3.3 is obtained step by step, covering the interval[0, vmax] with a finite number of steps of width vmin.

We now prove crucial estimates under the fundamental assumption that D(v)(the diffusion coefficient) does not depend on S. Such estimates are based on thepreviously found L∞ estimate for n. We stress the fact that our global existencetheorem is based particularly on this restrictive assumption, which we hope to dropin future papers.

To this purpose we need to introduce the next extra assumptions:

(B.1) ∂SV ∈ L∞loc as a function of v and S.

(B.2) nin ∈ C0([0, vmax];W

2∞(Ω)

).

(B.3) φ ∈ C0([0, vmax];W

2,1∞ (∂Ω× [0, T ])

).

From now on we make use of some special symbols for classical functional spacesand their norms, whose precise definitions are collected in a dedicated appendix atthe end of the paper.

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Let us start by proving the following proposition.Proposition 3.5. Under assumptions A and B, we have that any classical solu-

tion of our problem satisfies

(3.12) |n|(2−(N+2)/q)ΩT

+ ‖n‖(2)q,ΩT≤ M(q) ∀v ∈ [0, vmax], ∀q ≥ N + 2.

Proof. We first rewrite our problem in the form

∂tn−D(v)∇2n+ [n ∂SV · ∇S + V · ∇n− L(n)] = 0,(3.13)

n(x, v, 0) = nin(x, v),(3.14)

n(x, v, t) = φ(x, v, t), x ∈ ∂Ω. t ∈ [0, T ].(3.15)

Notice that assumptions A ensure that n is bounded. Moreover, assumptions Bare needed to apply, as we are going to do in a moment, Theorem 9.1 of [14] (seeAppendix B at the end of the present paper); indeed, (B.2) and (B.3) ensure thenecessary regularity of the boundary data, while (B.1) allows us to bound the sourceterm by means of a suitable norm of ∇n.

If we regard (3.13)–(3.15) as a linear problem, treating the terms in square brack-ets as known terms, we can apply Theorem 9.1 of [14, p. 341], thus obtaining

(3.16) ‖n‖(2)q,ΩT≤ C0

⎛⎝1 +

(∫ T

0

(‖∇n‖L∞(Ωt×[0,vmax]

)qdt

)1/q⎞⎠ ∀v ∈ [0, vmax],

where the integral in parentheses comes out by bounding from above the terms ∇Sand V · ∇n in the square brackets in (3.13). We recall that Ω is a bounded domainso that, in applying Theorem 9.1 quoted above, the Lq-norm is controlled by theL∞-norm. On the other hand, setting

(3.17) y(T ) := ‖∇n‖L∞([0,vmax]×ΩT )

for q > N+2, using (3.16) together with Theorem 9.1 and its corollary on pp. 341–342in [14], jointly with the corollary in Appendix B, we get

(3.18) y(T ) ≤ C1

⎛⎝1 +

(∫ T

0

yq(t)dt

)1/q⎞⎠ .

Setting yq(T ) := Z(T ) we have

(3.19) 0 ≤ Z(T ) ≤ C2

(1 +

∫ T

0

Z(t)dt

),

which by means of Gronwall’s lemma yields an estimate for y(T ) and hence because

of (3.16) for ‖n‖(2)q,ΩTfor all v ∈ [0, vmax], which in turn implies (see the corollary on

p. 342 of [14] and the corollary in Appendix B)

(3.20) |n|(2−(N+2)/q)ΩT

+ ‖n‖(2)q,ΩT≤ M(q, T ) ∀v ∈ [0, vmax], q > N + 2.

Remark 2. We emphasize that M(q, T ) is indeed a given constant since it de-pends on known quantities, For this reason, from now on the dependence on T in thebounding constants of the a priori estimates will be dropped.

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Recall that C0, C1, C2 depend on q but not on v.

Remark 3. In all the previous estimates we have strongly made use of the resultsof Proposition 3.3 (i.e., the boundedness of n).

At this point, due to (3.20), we regard the square bracket term in (3.13) as aknown term in Hα,α/2; thus applying Theorem 5.2 of [14, p. 320], under the followingextra assumptions, we get Proposition 3.6:

(C.1) nin ∈ C0([0, vmax];H2+α(Ω)).

(C.2) φ ∈ C0([0, vmax];H

2+α,1+α/2(∂Ω× [0, T ])).

(C.3) V is a C1+α function of S, uniformly with respect to v, ∂SQ ∈ L∞loc as a

function of v, w, S.(C.4) The second-order compatibility conditions hold true. (This assumption can

be dropped if we require a solution which has less regularity on ∂Ω× {T = 0}.)Proposition 3.6. Under assumptions A, B, and C we have

(3.21) |n|(2+α)ΩT

≤ M ∀v ∈ [0, vmax].

We recall that in all the estimates the constants M are independent of v ∈[0, vmax].

At this point we need to prove the final a priori estimate ensuring that

n ∈ C0([0, vmax], C2,1(ΩT )).

To this purpose we have to stipulate some extra assumptions which can be dropped(see the next remark) if we look for a solution which only belongs to L∞([0, vmax],C2,1(ΩT )). For this reason, for simplicity we prove a somehow stronger result undereven more restrictive assumptions.

Let us assume the following:

(P.1) D, Q, have their derivatives with respect to v locally bounded w.r.t. all theirvariables, ∂vV is a C1+α function of S, uniformly w.r.t v.

(P.2) ∂vnin ∈ W 1∞([0, vmax];H

2+α(Ω)).(P.3) ∂vφ ∈ W 1

∞[0, vmax];H2+α,1+α/2(∂Ω× [0, T ])).

We now differentiate system (2.1)–(2.2)–(2.3) with respect to v and put z = ∂vn.Using estimates (3.21) we can apply Theorem 9.1 of [14], thus obtaining that underassumptions A, B, C, and P,

(3.22) ‖z‖(2)q,ΩT≤ C0

⎛⎝1 +

(∫ T

0

(‖z‖L∞(Ωt×[0,vmax]

)qdt

)1/q⎞⎠ ∀v ∈ [0, vmax].

At this point, working as in Proposition 3.5, we have

(3.23) |z|(2−(N+2)/q)ΩT

+ ‖z‖(2)q,ΩT≤ M(q) ∀v ∈ [0, vmax], q > N + 2.

Finally, with the usual bootstrap argument, we obtain

(3.24) |z|(2+α)ΩT

≤ M ∀v ∈ [0, vmax],

which is the estimate we were looking for.

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Remark 4. Note that by means of a regularization procedure with respect tov, we can always assume that hypotheses P are satisfied. Then, by passing to thelimit in the relaxation parameter and bearing in mind that the estimates found insections 1, 2, and 3 will not depend on it, we obtain a solution of our problem whichhas only the regularity appearing in Proposition 3.5.

4. Local existence and uniqueness for the case of Dirichlet boundaryconditions. Here local existence will be achieved in a small time interval (0, τ) via acontraction mapping argument applied to (2.1) coupled with (2.2)–(2.3). (This localexistence theorem will be proved in the general case in which D depends also on S.)The following extra assumptions are needed for V , Q,D. There exists a finite positiveconstant M such that(D.1) |∂SV S |, |∂2

SSV | ≤ M ∀v ∈ [0, vmax] in Ω× [0, T ];(D.2) |∂SQ| ≤ M ∀v ∈ [0, vmax] in Ω× [0, T ];(D.3) |∂SD|, |∂2

SSD| ≤ M ∀v ∈ [0, vmax] in Ω× [0, T ].Now we put (2.1) in nondivergence form,

(4.1) ∂tn− [D]∇2n+ [V ] · ∇n+ [R] = [L(n; v)],

where we set

R := (∂SV ) · (∇S)n− (∂SD)(∇S) · (∇n),

and we append to it the following initial and boundary conditions:

(4.2) n(x, v, 0) = nin, x ∈ Ω,

(4.3) n = φ, (x, t) ∈ ∂Ω× [0, T ].

Then we define the Banach space

H ={n ∈ L∞

([0, vmax];H

1+α,1/2+α/2(Ωτ ))| ‖|n ‖| ≤ M

},

where M is a positive constant to be determined later, ‖|•‖| denotes the norm ofL∞([0, vmax];H

1+α,1/2+α/2(Ω× [0, τ ])), and the map

T : H �→ His defined as

T (n) = n,

where n is the solution of the linear problem (4.1)–(4.3) in the time interval [0, τ ],provided that n replaces n in all terms in square brackets. (Recall that n appears alsoin the definition of S.) Applying Theorem 5.3 in [14, p. 320] and under assumptionsA to D, we get

(4.4) ‖n‖L∞([0,vmax];H2+α,1+α/2(Ωτ )) ≤ F (M),

where F is a positive increasing function of M and, in principle, also of τ . Since τ ≤ Twith T fixed, here and in what follows this dependence is not explicitly emphasized.Hence we have

(4.5) ‖|n ‖| ≤ C0(1 + F (M)τα/2),

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where C0 does not depend on M , which implies that fixing M sufficiently large andτ sufficiently small we get

‖| n‖| ≤ M,

i.e., T ⊂ H .At this point, in the time interval [0, τ ], inequality (4.5) can be regarded as an a

priori estimate which, in fact, will be used to bound some crucial coefficients in theforthcoming equation (4.7). It remains to prove that T is a contraction mapping. Tosimplify notation we set

(4.6) T (n2)− T (n1) = n2 − n1 := �, n2 − n1 := �.

By a standard “adding-subtracting” technique it is easy to see that � obeys thefollowing set of equations:

∂t�− [D1]∇2�+ [V 1] · ∇� = ∇2n2([D2]− [D1])−∇n2 · ([V 2]− [V 1]),(4.7)

+ [R1]− [R2] + [L2]− [L1] := G,

(4.8) �(x, v, 0) = 0,

(4.9) �(x, v, t)|∂Ω = 0.

Here [Dj ], [V j ], [Rj ], [Lj ] denote the corresponding quantities evaluated for n =nj . Thus we regard all terms in (4.7) as known ones. In particular, thanks to (4.4),the function G in (4.7) is L∞-bounded in terms of the ‖|•‖| of the entries. Thus wecan apply Theorem 9.1 of [14], yielding

(4.10) ‖�|(2)q,Ω×(0,τ) ≤ c(‖G‖q,Ω×(0,τ)

) ≤ c|Ω|1/qτ1/q‖G‖∞,Ω×(0,τ),

that is,

(4.11) ‖�|(2)q,Ω×(0,τ) ≤ τ1/qC(M, q) ‖|�‖|.At this point we apply the corollary in Appendix B, which gives the Holder

continuity of ∇� in terms of estimate (4.11), provided that q > N+21−α , and hence

(4.12) ‖|�‖| ≤ τ1/qC(M, q) ‖|�‖|, q >N + 2

1− α.

Being M and q fixed, C is also a fixed constant and so we are free to choose τ insuch a way that C(M, q)τ1/q < 1 so that T turns out to be a contraction.

This implies local existence and uniqueness for our problem. Then the a prioriestimates of the previous section allow us to iterate such a local existence procedure,thus obtaining a solution which is global in time, i.e., defined in the same time intervalin which the coefficients of the PDE and the boundary data are defined. We havethus proved the following first main theorem of our paper.

Theorem 4.1. Under the regularity assumptions A to D, problem (2.1) with theinitial and boundary conditions (2.2)–(2.3) admits one and only one classical solutionsuch that

n ∈ L∞([0, vmax];H

2+α,1+α/2(ΩT )).

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Remark 5. Hypotheses C and D may somehow be weakened but at the expenseof brevity. The advantage here in using these assumptions is the possibility to useclassical regularity theorems.

Remark 6. Once a solution with the regularity as in Theorem 4.1 has beenobtained, proceeding as at the end of the previous section, we get that the solutionalso belongs to W 1

∞([0, vmax];H

2+α,1+α/2(ΩT )). Naturally, if only assumptions A to

D hold, working as sketched in Remark 4, we get that only the regularity of Theorem4.1 is obtained.

Remark 7. One may wonder what happens to estimate (4.12) when α → 1. If thisis the case, the assumptions on the regularity of the boundary data hold uniformly asα → 1 (which means replacing therein H2+α with W 3

∞). However, the regularity ofthe solution remains an H2+α-regularity for all α ∈ (0, 1) and one does not achieveany W 3

∞-regularity for the solution, since some estimates, for example, (4.12), couldblow up as α → 1 due possibly to the divergence of C(M, q).

5. The case of Neumann boundary conditions: a priori estimates. Weconsider again the evolution problem (2.1) but now under Neumann boundary condi-tions. Of course different techniques will be needed; nevertheless some of the previousresults still hold with minor modifications.

Equation (2.1) will be complemented with the initial and boundary conditions

(5.1) n(x, v, 0) = nin(x, v) , x ∈ Ω,

(5.2) j(x, v, t) · ν = α(S, v, t) (n∗ − n(x, v, t)) , x ∈ ∂Ω, t ∈ [0, T ],

where ν is the inward unit normal to ∂Ω and α, n∗ can vary on different connectedcomponents of ∂Ω. In (5.1) it is tacitly assumed that nin does not vanish identicallyin Υvmax .

Assuming hypotheses A and(E) 0 ≤ α ≤ M and n∗ > 0,

we have that Proposition 3.1 continues to hold without any change, as is quite obviouschecking its proof.

Proposition 3.2 also holds unchanged in conclusions but the proof needs somemodifications and is fully presented for clarity.

Proposition 5.1. Under assumptions A and E, for all T > 0 there existM(T, v) > 0 and v(T ) ≤ vmax s.t. |n(x, t, v)| ≤ M(T, v) in ΘT,v.

Proof. As a preliminary step let us observe that for a sufficiently small T1 we have

(5.3) maxΘT1,vmax

|n(x, t, v)| ≤ 2I,

where I := maxΥvmax|nin(x, v)|, where obviously T1 is not controlled from below (see

Proposition 3.2).We have that 0 ≤ n ≤ n, where n solves

(5.4) ∂tn+∇ · ([V (n)]n− [D(n)]∇n) =1

2

∫ v

0

Q(v −w,w)n(v −w)n(w)dw := [L]

with the same initial-boundary condition as (2.1) and

j(x, v, t) · ν = [α] (n∗ − n(x, v, t)) , x ∈ ∂Ω.

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The symbol [f ] is used to recall that function f , whatever it is, is meant to be evaluatedon n, not on n. Notice that [L] ≥ [L]. In this regard (5.4) together with its boundaryconditions has to be read as a linear equation in n, the terms V , D, α and theoperator L of (5.4) being viewed as calculated on n which, at this step, is supposedto be known.

Let us now define M(T, v) := maxΘT,v|n(x, t, v)|. Obviously

(5.5) L(n; v) ≤ QmaxM2v

if v ∈ [0, v].We now proceed as in Theorem 7.1 of [14, p. 181], i.e., we multiply (5.4) by

n(k) := max(n− k, 0) (k ∈ �) and we get, because of the assumptions,

1

2

∫Ω

[n(k)(x, t)]2 |t=τt=0 dx+

∫ τ

0

{∫Ω

(D[n]∇n− V [n]n) · ∇n(k))dx

}dt(5.6)

−∫ τ

0

n(k)(j(x, v, t) · ν)|∂Ωdt =

∫Ωτ

Ln(k)ds.

If k > max(n∗, I) := k, we have, because of the assumptions, that the third integralis negative and can be dropped. Moreover∫

Ω

[n(k)(x, t)]2 |t=0 dx = 0.

Then, by an easy calculation and taking the supremum with τ ∈ [0, t1], we get

1

2sup

τ∈[0,t1]

∫Ω

[n(k)(x, τ)]2dx+1

2K

∫ t1

0

∫Ω

(∇n(k))2dsdt(5.7)

≤∫ t1

0

∫Ak(t)

[(1

2K2 +QmaxM

2v

)(n(k)(x, τ)

)2+ k2

]dsdt,

where Ak(t) = {x ∈ Ω | n(x, t) > k}.Thus by setting

D =

(1

2K2 +QmaxM

2v

),

we have obtained an inequality like the one following inequality (7.7) of [14, p. 184].We notice explicitly that D is bounded.

Such inequality implies an L∞-estimate of n (remember, in this regard,Theorem 6.1 of [14, p. 102] and Remark 6.2 [14, p. 103]). It is important to stress

the fact that this estimate depends on QmaxM2v but, since (5.4) (jointly with the

appropriate initial and boundary conditions) is regarded as a linear problem (D, V ,and α being bounded quantities), we get that the estimate depends linearly on the

quantity QmaxM2v as a whole, i.e.,

0 ≤ n ≤ n ≤ C0 + C1QmaxM2v ∀v ∈ [0, v],

which in turn implies

(5.8) M ≤ C0 + C1QmaxM2v.

From here on the proof continues as in Proposition 3.2.

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Proposition 3.3 remains true also in the case of Neumann boundary conditions,with minor modifications in the proof.

As far as Proposition 3.4 is concerned, it can be stated under some restrictionson the boundary flux.

Proposition 5.2. Let n(x, v, 0) ≡ 0 for all (x, v) ∈ ΘT,vmin and n∗ = 0. Thenn(x, v, t) ≡ 0 for all x ∈ Ω and v ∈ (0, vmin).

The proof is quite similar to that of Proposition 3.4 and is omitted. We justnotice that assuming n∗ = 0 allows us to drop the positive term (in the integrationby parts) coming out from the lateral boundary.

We now rewrite our problem in the form (recall that we still continue to take Dto be independent of S)

(5.9) ∂tn−D(v)∇2n = − [n ∂SV · ∇S + V · ∇n− L(n)] ,

(5.10) n(x, v, 0) = nin(x, v), x ∈ Ω,

(5.11) (∇n(x, t, v) · ν) |∂Ω×[0,T ]

= [B(S, n, v)], (x, t) ∈ ∂Ω× [0, T ],

where B = (α/D)(n− n∗) + (V · ν/D), and state the following assumption:(F.1) |∂tα|, |∂Sα|≤ M , for all v ∈ [0, vmax].(F.2) First-order compatibility condition is satisfied.Here, terms in square brackets are regarded as known and, consequently, we regard

(5.9)–(5.11) as a linear problem to which Theorem 9.1 of [14, p. 341] is applicable.Such theorem holds with the relevant changes also for the case of Neumann boundaryconditions. (See the remark at the end of the proof of the cited reference.) In thiscase the lateral boundary conditions must have the ‖•‖1−1/q,1/2−1/(2q)

q,∂ΩT-norm bounded.

Hence, under assumptions A, B, E, F, we get

(5.12) ‖n‖(2)q,ΩT≤ C0

⎛⎝1 +

(∫ T

0

(‖n‖L∞([0,vmax],C1,1/2(Ωt))

)qdt

)1/q⎞⎠ ,

where the integral on the left-hand side comes by bounding from above the terms insquare brackets in (5.9), (5.11).

At this point we can conclude as in Proposition 3.5, thus obtaining

(5.13) |n|(2+α)ΩT

≤ M ∀v ∈ [0, vmax]

(being now y(T ) := ‖n‖L∞([0,vmax],C1,1/2(ΩT ))).We recall again that in all the estimates the constants M are independent of

v ∈ [0, vmax].Finally, proceeding as at the end of section 3, we get estimates like (3.23) and

(3.24), provided that the extra assumptions P.1 and P.2 hold together with thefollowing:

(P.4) ∂vα is a C1+α function of S, t, uniformly w.r.t. v.

6. Local existence and uniqueness for the case of Neumann boundaryconditions. The argument is based again on a contraction mapping technique ap-plied to (2.1) and (5.1)–(5.2). The steps needed are about the same as in section 4;for clarity we rephrase the full path leading to the conclusion.

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The following extra regularity assumptions are needed for α. There exists a finitepositive constant M such that the following holds:

(G) |∂SSα|, |∂2ttα|, |∂2

tSα| ≤ M for all v ∈ [0, vmax] and (�x, t) ∈ Ω× [0, T ].Now we put (2.1) in nondivergence form:

(6.1) ∂tn− [D]∇2n+ [V ] · ∇n+ [R] = [L(n; v)],

where we set

R := (∂SV · ∇S)n− ∂SD(∇S · ∇n),

and we append to it the initial and boundary conditions (5.10)–(5.11).Then we define the Banach space

H ={n ∈ L∞

([0, vmax];H

1+α,1/2+α/2(Ω× [0, τ ]))|‖|n‖| ≤ M

},

where M is a positive constant to be determined later, and the map

T : H �→ H

is defined as

T (n) = n,

where n is the solution of the linear problem (5.10), (5.11), (6.1) in the time interval[0, τ ], provided that n replaces n in all terms in square brackets.

Under the assumptions A to G, Theorem 5.3 of [14] can be applied, and we get

(6.2) ‖n‖L∞([0,vmax];H2+α,1+α/2(Ω×[0,τ ])) ≤ F (M),

where F is a positive increasing function. Inequality (6.2) implies

‖|n‖| ≤ C0(1 + F (M)τα/2),

where C0 does not depend on M , which implies that fixing M sufficiently large andτ sufficiently small we get

‖|n‖| ≤ M,

i.e., T (H ) ⊂ H .It remains to prove that T is a contraction mapping. Using again notation (4.6)

and the usual adding-subtracting technique, it is easy to see that � obeys the followingset of equations:

∂t�− [D1]∇2�+ [V 1] · ∇� = ∇2n2 · ([D2]− [D1])−∇n2([V 2]− [V 1])(6.3)

+ [R1]− [R2] + [L2]− [L1],

(6.4) �(x, v, 0) = 0,

(6.5) ∇� · ν = [B2]− [B1].

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As before, all quantities in square brackets denote the corresponding quantitiesevaluated for n = nj . Thus we regard [D1] and [V 1] together with all right-hand-sideterms in (6.3) to (6.5) as known terms. Using (6.2) together with Theorem 9.1 of [11]and the corollary in Appendix B, it follows that

(6.6) ‖|�‖| ≤ τ1/qC(M, q) ‖|�‖|, q >N + 2

1− α.

For fixed q as in (6.6), we choose τ satisfying C(M, q)τ1/q < 1 so that T turnsout to be a contraction. This implies local existence and uniqueness for the wholeproblem.

At this point the a priori estimates of the previous section allow us to iteratesuch local existence procedure, thus obtaining a solution which is global in time,i.e., defined in the same time interval in which the coefficients of the PDE and theboundary data are defined. We have thus proved the second main theorem of ourpaper, as follows.

Theorem 6.1. Under assumptions A to G, problem (2.1) with the initial andboundary conditions (5.1)–(5.2) admits one and only one classical solution such that

n ∈ L∞([0, vmax];H

2+α,1+α/2(Ω× [0, T ])).

Of course a remark similar to Remarks 5 and 6 can be stated.

7. Concluding remarks. The main result of this paper is the proof of the exis-tence and uniqueness of a regular solution, globally in time, of the model representedby 2.1 coupled with both Dirichlet as well as Neumann boundary data. The mostimportant restriction needed to achieve this goal is mainly the independence of thediffusion coefficient D on concentration S. On the contrary, our result can be gen-eralized in various directions (even though they have not been explicitly stated, forbrevity). We will briefly describe them here. First, it is possible to assume thatD,V, Q depend on x and t, provided that sufficient regularity assumptions are made.For example, it suffices to assume that D,V, Q and all their derivatives appearing inthe statements of the theorems presented in this paper belong to H2+α,1+α/2(ΩT ),uniformly with respect to S and v. Also, we can drop the compatibility conditions(with the exception of that of order zero); obviously, in this case, the solution will notbe regular on ∂Ω× {t = 0}.

A final comment concerns the question of whether the concentration functionS(x, t) remains always in [0, 1]. We are not able to prove it if both D and V depend,besides (S,x, t), on v. However, if we multiply (2.1) by v and integrate it with respectto v on [0, vmax], the following parabolic equation can be easily achieved:

∂S

∂t−D∇2S=

(∇xD +

∂D

∂S∇S −V

)· ∇S − S∇ ·V,

where

∇ ·V = ∇x ·V +∂V

∂S· ∇S.

We notice that the vanishing of the integral containing the coalescence operatorLc is justified by its well-known property of being volume preserving (see, e.g., [6]).Then, under the less general assumptions D = D(S,x, t), V = V(S,x, t) and the

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constraint ∇x · V ≥ 0 (satisfied, for example, if V does not depend explicitly onx), a suitable choice of the boundary data, the maximum principle, and the parabolicversion of Hopf’s lemma (see, respectively, Theorem 4 and Theorem 14 of [10]) ensuresthat 0 ≤ S(x, t) ≤ 1.

Appendix A. We collect here some quite standard notation and some other,less common notation taken from [14]. Let Ω be an open bounded subset of �N withLipschitz boundary.

For functions of (x, t) ∈ ΩT , we make use of the following notation:1. |f |C0(ΩT ) is the standard supremum norm for f continuous in ΩT , where

C0(ΩT ) is the corresponding Banach space.2. ‖f‖Lq(ΩT ) is the standard norm for q-powered integrable functions in ΩT ,

where Lq(ΩT ) is the corresponding Banach space.3. W 2,1

q (ΩT ) is the Banach space of functions having the following norm bounded:

‖f‖(2)q,ΩT= ‖f‖Lq(ΩT ) + ‖∂tf‖Lq(ΩT ) + ‖∇f‖Lq(ΩT ) + ‖D2

xf‖Lq(ΩT ).

4. C2,1(ΩT ) is the Banach space of functions having the following norm bounded:

|f |(2)ΩT= |f |C0(ΩT ) + |∂tf |C0(ΩT ) + |∇f |C0(ΩT ) + |D2

xf |C0(ΩT ).

5. Following [14] we set

〈f〉(α/2)t,ΩT:= sup

t′ �=t′′

|f(x, t′)− f(x, t′′)||t′ − t′′|α/2

and

〈f〉(α)x,ΩT:= sup

x′ �=x′′

|f(x′, t)− f(x′′, t)||x′ − x′′|α ;

then, for f ∈ C0(ΩT ) and α ∈ (0, 1), |f |(α)ΩTdenotes the norm

|f |(α)ΩT:= |f |(0)ΩT

+ 〈f〉(α)x,ΩT+ 〈f〉(α/2)t,ΩT

,

where Hα,α/2(ΩT ) the corresponding Banach space of functions with |f |(α)ΩTbounded.

6. C1,1/2(ΩT ) is the Banach space of functions having the following norm bounded:

|f |C0(ΩT ) + |∇f |C0(ΩT ) + 〈f〉(1/2)t,ΩT.

7. The spaces H1+α,(1+α)/2(ΩT ) and H2+α,1+α/2(ΩT ) are defined accordingly:

|f |(1+α)ΩT

:= |f |(0)ΩT+ 〈f〉(1+α)/2

t,ΩT+ |∇f |(α)ΩT

,

|f |(2+α)ΩT

:= |f |(1+α)ΩT

+ |∂tf |(α)ΩT+ |D2

xf |(α)ΩT,

respectively.

Appendix B. Here we recall explicitly some results, proved in [14] and [11], usedextensively throughout this paper. When possible, we state the relevant results in asimpler and less general form, more suitable for the use we made here.

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In what follow L identifies the linear parabolic operator

(B.1) L ≡ ∂tu−Mu− f,

where

(B.2) M =∂

∂xi[aij(x, t)uxi + ai(x, t)u]− bi(x, t)uxi − a(x, t)u

with discontinuous and, generally, nondifferentiable and unbounded coefficients satis-fying the condition of uniform parabolicity, namely,

νn∑

i=1

ξ2i ≤ aij(x, t)ξiξj ≤ μn∑

i=1

ξ2i , ν, μ = const > 0.

Theorem 7.1 in [14, p. 181]. If q > N + 2, ai, bi ∈ L2q(ΩT ), a, f ∈ Lq(ΩT ),

with norms bounded by μ1, then for any weak solution of L(u) = 0 not exceeding k

on ∂pΩT , supΩTu is finite and estimated by N, k, μ1, q and the constants of uniform

parabolicity.In the statement above, ∂pΩT denotes the usual parabolic boundary, i.e., the set

(∂Ω× [0, T ]) ∪ (Ω× {t = 0}) .The proof of Theorem 7.1 relies on a particular estimate for the norm

|u|ΩT := sup0≤t≤T

‖u‖L2(Ω) + ‖∇u‖L2(Ω).

Such estimate follows from inequality (7.7) on p. 184 of [14] appearing in the proofof the previously quoted Theorem 7.1. The same estimate also can be obtained in thecase of Neumann boundary conditions, provided that the condition k > max(n∗, I) issatisfied. Such an estimate enables us to use the following result of [14], thus obtainingthe a priori estimate of supΩT

u.

Theorem 6.1 in [14, p. 102]. Suppose that sup∂Ω×[0,T ] u ≤ k, k ≥ 0, and that theinequalities

(B.3) |u|Ω×[0,T ] ≤ γK [μ(k)](1+κ)/r

hold for k ≥ k with certain positive constants γ and κ. Here μ(k) =∫ T

0mesr/qAk(t) dt

and Ak(t) is defined as at the end of formula (5.7), while q and r are arbitrary numberssatisfying the constraint

1

r+

N

2q=

N

4, q, r > 2.

Then

supΩ[0,T ]

u ≤ M,

where M depends on all the constants in the statement.Furthermore, Remark 6.2 in [14] just states that “under the conditions of theorem

6.1, it is possible to drop the requirement of boundedness for sup∂Ω×[0,T ] u and to

assume that inequalities (B.3) are fulfilled for k larger than a certain k ≥ 0.”

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Consider now the problem

(B.4)

⎧⎨⎩Lu = f(x, t),

u|t=0 = φ(x), u|∂Ω×[0,T ] = Φ(x, t),

with

L

(x, t,

∂

∂t,∂

∂x

)u =

∂u

∂t−

N∑i,j=1

aij(x, t)∂2 u

∂xi∂xj+

N∑i=1

ai(x, t)∂u

∂xi+ a(x, t)u.

Theorem 5.2 in [14, p. 320]. Suppose that α ∈ (0, 1), the coefficients of theoperator L belong to the class Hα,α/2(ΩT ), and the boundary ∂Ω belongs to the classHα+2. Then for any f ∈ Hα,α/2(ΩT ), φ ∈ Hα+2(Ω), Φ ∈ Hα+2,α/2+1(∂Ω × [0, T ])satisfying the compatibility conditions of order 2, problem (B.4) has a unique solutionfrom the class Hα+2,α/2+1(∂Ω× [0, T ]) and it satisfies the inequality

(B.5) |u|(α+2)ΩT

≤ c(|f |(α)ΩT

+ |φ|(α+2)Ω + |Φ|(α+2)

∂Ω×[0,T ]

).

Consider finally the problem

(B.6)

⎧⎪⎪⎪⎨⎪⎪⎪⎩Lu = f(x, t),

u|t=0 = φ(x),

B

(x, t,

∂

∂x

)u|∂Ω×[0,T ] = Φ(x, t),

with

B

(x, t

∂

∂x

)u|∂Ω×[0,T ] =

N∑i=1

bi(x, t)∂u

∂xi+ b(x, t)u|∂Ω×[0,T ].

Theorem 5.3 in [14, p. 320]. Suppose that α ∈ (0, 1), the coefficients of theoperator L belong to the class Hα,α/2(Ω× [0, T ]), and finally, bi, b ∈ Hα+1,α/2+1/2(Ω×[0, T ]). Then for any f ∈ Hα,α/2(Ω× [0, T ]), φ ∈ Hα+2(Ω), Φ ∈ Hα+1,(α+1)/2(∂Ω×[0, T ]) satisfying the compatibility conditions of order 1, problem (B.6) has a uniquesolution from the class Hα+2,α/2+1(Ω× [0, T ]) with

(B.7) |u|(α+2)ΩT

≤ c(|f |(α)ΩT

+ |φ|(α+2)Ω + |Φ|(α+1)

∂Ω×[0,T ]

)Theorem 9.1 in [14, p. 341]. Let q > N + 2; suppose that the coefficients aij of

the operator L are continuous functions in Ω × [0, T ], while the coefficients ai and ahave finite norms ‖ • ‖q,ΩT . Then for any f ∈ Lq(Ω × (0, T )), φ ∈ W 2−2/q

q (Ω) and

Φ ∈ W 2−1/q,1−1/(2q)q (∂Ω× (0, T ) satisfying the compatibility condition of zero order

φ∂Ω = Φt=0,

problem (B.4) has a unique solution u ∈ W 2,1q (Ω× (0, T )) which satisfies the estimate

(B.8) ‖u‖(2)q,Ω×(0,T ) ≤ c(‖f‖q,Ω×(0,T ) + ‖φ‖(2−2/q)

q,Ω + ‖Φ‖(2−1/q)q,∂Ω×(0,T )

).

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We remark that the last two norms in the right-hand side of (B.8) are the standardones in the fractional Sobolev spaces appearing in the statement of Theorem 9.1.

Corollary. If the conditions of Theorem 9.1 are fulfilled for q > (N + 2)/2,then the solution of problem (B.4) and its derivatives with respect to the space variablessatisfy a Holder condition in x and t, namely,

|u|2−(N+2)/qΩT

≤ c(T )(‖u‖(2)q,ΩT

).

We notice explicitly that the constant c(T ) in the inequality above blows up asT → 0. This drawback can be superseded proceeding as in [11, Appendix 2], wherethe following embedding result is proved:

(B.9) |u|2−(N+2)/qΩ×[0,T ] ≤ M

(‖u‖(2)q,Ω×[0,T ] + ‖u(x, 0)‖|2−2/q

Ω

),

valid for any q > (N + 2)/2, q �= N + 2, and with M remaining bounded forbounded T .

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[2] H. Amann and C. Walker, Local and global strong solutions to continuous coagulation-fragmentation equations with diffusion, J. Differential Equations, 218 (2005), pp. 159–186.

[3] H. Amann and F. Weber, On a quasilinear coagulation-fragmentation model with diffusion,Adv. Math. Sci. Appl., 11 (2001), pp. 227–263.

[4] P. B. Dubovskii and I. W. Stewart, Existence, uniqueness, and mass conservationfor the coagulation–fragmentation equation, Math. Methods Appl. Sci., 19 (1996),pp. 571–591.

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MODELING DEGRADATING DISPERSIONS IN A THREE-DIMENSIONAL FINITE CONTAINER UNDER GENERAL BOUNDARY...

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