The process of fragmentation arises in many physical situations, including polymer degradation, droplet breakage and rock crushing and grinding. Under suitable assumptions, the evolution of the size distribution c(x, t), where x represents particle size and t is time, may be described by the linear integro-differential equation

Blair, P.N.

Lamb, W.

Stewart, Iain W.

English

University of Strathclyde Institutional Repository

Strathprints Institutional RepositoryBlair, P.N. and Lamb, W. and Stewart, Iain W. (2004) An application of semigroup theory to afragmentation equation. In: Integral Methods in Science and Engineering. Birkhauser, Boston,USA, pp. 19-24. ISBN 081764377XStrathprints is designed to allow users to access the research output of the University of Strathclyde.Copyright c© and Moral Rights for the papers on this site are retained by the individual authorsand/or other copyright owners. You may not engage in further distribution of the material for anyprofitmaking activities or any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or study, educational, ornot-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to Strathprints administrator:mailto:strathprints@strath.ac.ukhttp://strathprints.strath.ac.uk/An Application of SemigroupTheory to a FragmentationEquationP.N. Blair, W. Lamb and I.W. Stewart1. IntroductionThe process of fragmentation arises in many physical situations, includingpolymer degradation, droplet breakage and rock crushing and grinding.Under suitable assumptions, the evolution of the size distribution c(x, t),where x represents particle size and t is time, may be described by thelinear integro-differential equation∂c(x, t)∂t= −a(x)c(x, t) +∫ ∞xa(y)b(x, y)c(y, t)dy, x > 0, t > 0 ; (1)see [1] for details. In (1), a(x) describes the rate at which a particle ofsize x fragments and b(x, y) represents the rate of production of particlesof size x due to the break-up of particles of size y > x. For the total massin the system to remain constant during fragmentation, b must satisfy thecondition ∫ y0x b(x, y) dx = y .Here we consider the casea(x) = xα+1 , b(x, y) = (ν + 2)(x/y)ν y−1 , (2)where α ∈ R and −2 < ν ≤ 0 . The restriction ν > −2 is mathematicalto ensure integrability whereas ν ≤ 0 is a physical property of the systemthat is explained in [1]. With these rate functions, equation (1) becomes∂c(x, t)∂t= −xα+1c(x, t)+(ν+2)∫ ∞x(x/y)ν yα c(y, t)dy, x > 0, t > 0 . (3)Our aim is to show that the questions of existence, uniqueness and massconservation of solutions to (3) can be resolved by using the theory ofThe work of the first author is supported by a Scholarship from the CarnegieTrust for the Universities of Scotland.2 P.N. Blair, W. Lamb, I.W. Stewartsemigroups of linear operators. For certain values of the parameters, theequation can be dealt with immediately by simply quoting previously estab-lished results. To cater for the remaining values, we use the idea of similarsemigroups [2, p.43] to determine the required information on solutions. Inthe case when α > −1, we also obtain an explicit formula for the semigroupassociated with the equation. This yields a closed-form expression for thesolution that coincides with that given in [1].2. The Abstract ProblemBefore semigroup theory can be applied, the initial-value problem asso-ciated with equation (3) is reformulated as an abstract Cauchy problem(ACP) of the formddtc(t) = Aν,α[c(t)] , t > 0 , c(0) = c0, (4)where the operator Aν,α is defined on suitable functions f by(Aν,αf)(x) = −xα+1f(x) + (ν + 2)∫ ∞x(x/y)ν yα f(y) dy, x > 0 .As solutions to (3) are expected to be mass-conserving, an appropriateBanach space in which to study (4) isY = {f : ||f || =∫ ∞0x|f(x)| dx < ∞} .In this case, a natural domain for Aν,α isD(Aν,α) = {f ∈ Y : xα+1f ∈ Y } ,since a routine calculation shows that Aν,αf ∈ Y whenever f ∈ D(Aν,α) .We note that the solution of (4) is interpreted as a strongly continu-ously differentiable Y -valued function of t. Such a solution is sought in theform c(t) = Sν,α(t)c0, t ≥ 0, where {Sν,α(t)}t≥0 is a strongly continuoussemigroup of linear operators on Y that is generated by some extension of(Aν,α, D(Aν,α)). This semigroup is said to be substochastic if Sν,α(t) ≥ 0and ||Sν,α(t)|| ≤ 1 for each t ≥ 0. If, additionally, ||Sν,α(t)f || = ||f || for allt ≥ 0 and f ∈ Y+, where Y+ is the cone of non-negative a.e. functions inY , then {Sν,α(t)}t≥0 is called a stochastic semigroup.A Fragmentation Equation 33. Existing ResultsFor the case α = −1, the operator Aν,−1 is bounded on Y and hence is theinfinitesimal generator of a uniformly continuous semigroup {Sν,−1(t)}t≥0on Y , whereSν,−1(t) =∞∑k=0Akν,−1tk/k! , t ≥ 0 .As shown in [3, Lemma 2.4], this semigroup is stochastic and this, togetherwith [2, p. 145, Proposition 6.2], establishes that the corresponding ACPhas a unique, non-negative, mass-conserving solution c(t) = Sν,−1(t)c0 forall t ≥ 0 and c0 ∈ Y+.When α > −1, the functions a and b defined by (2) satisfy the constraintsimposed in [4]. Consequently, in this case, the semigroup {Sν,α(t)}t≥0 canbe obtained by means of a truncation/limit procedure which takes the formS(n)ν,α(t) =∞∑k=0(Aν,αPn)ktk/k! = I − Pn + exp(Aν,αt)Pn , t ≥ 0 , (5)Sν,α(t)f = limn→∞S(n)ν,α(t)f , f ∈ Y, t ≥ 0 , (6)where(Pnf)(x) ={f(x), if 0 < x < n;0, if x ≥ n. (7)Theorem 1. The semigroup {Sν,α(t)}t≥0 defined via (5)–(7) is stochasticand is generated by an extension (A˜ν,α, D(A˜ν,α)) of (Aν,α, D(Aν,α)). Hencethe ACPddtc(t) = A˜ν,α[c(t)] , t > 0 , c(0) = c0 ,has a unique, non-negative, mass-conserving solution for each non-negativec0 ∈ D(A˜ν,α).Proof. See [4, Sections 2–5]One drawback to the approach used in [4] is that it cannot be appliedwhen α < −1 . However, in [5] an alternative strategy involving the Kato–Voigt perturbation theorem is used to analyse (4) for any α 6= −1 butunder the additional constraint that ν = 0 .Theorem 2. For each α 6= −1 there exists a smallest substochastic semi-group {S0,α(t)}t≥0 on Y generated by an extension of (A0,α, D(A0,α)).Furthermore, if α > −1, then this semigroup is stochastic with generator(A0,α, D(A0,α)) (i.e. the closure of (A0,α, D(A0,α))), whereas if α < −1,the generator is a proper extension of (A0,α, D(A0,α)). In the latter case,for each f ∈ Y+, there exists t > 0 such that ||S0,α(t)f || < ||f ||, i.e. a massloss occurs.Proof. See [5, Sections 4,5].4 P.N. Blair, W. Lamb, I.W. Stewart4. Extension of ResultsWe rely on the idea of similar semigroups to extend the results stated inTheorem 2 to the case when ν 6= 0. Motivated by transformations used in[1], we introduce an operator, Rν , defined on Y by(Rνf)(x) = x−2ν/(ν+2)f(x2/(ν+2)).It is a straightforward matter to show that Rν is a homeomorphism fromY onto Y . Moreover,RνAν,αR−1ν = A0,β , where β = (2α− ν)/(ν + 2) .Consequently, as pointed out in [1], it is possible to transform the originalmultiple fragmentation problem into an equivalent binary problem.Theorem 3. For each α 6= −1 and ν ∈ (−2, 0] there exists a smallestsubstochastic semigroup {Sν,α(t)}t≥0 on Y generated by an extension of(Aν,α, D(Aν,α)). Furthermore, if α > −1, then this semigroup is stochas-tic with generator (Aν,α, D(Aν,α)), whereas if α < −1, the generator is aproper extension of (Aν,α, D(Aν,α)). In the latter case, for each f ∈ Y+,there exists t > 0 such that ||Sν,α(t)f || < ||f ||, i.e. a mass loss occurs.Proof. Since Sν,α(t) = R−1ν S0,β(t)Rν ∀t ≥ 0, where β = (2α− ν)/(ν+2),and β > −1⇔ α > −1, the result follows from Theorem 2.Corollary 1. If α > −1 and ν ∈ (−2, 0] then the semigroups in Theorems1 and 3 are identical.Proof. This follows since the closure of the operator (Aν,α, D(Aν,α))(which exists from Theorem 3) can be shown to be the infinitesimal gener-ator of the semigroup defined by (5)–(7).5. Exact SolutionsFollowing an approach similar to that used in [3] for the case α = −1,we now obtain an explicit formula for the semigroup {Sν,α(t)}t≥0 whenα > −1. Again, it is convenient to apply a similarity transformation tosimplify Aν,α. Defining the operator Wν,α and Banach space Yν,α by(Wν,αf)(x) = x−ν/(α+1)f(x1/(α+1)),Yν,α = {f : ||f ||ν,α =∫ ∞0x(ν−α+1)/(α+1)|f(x)| dx <∞},it can be shown that Wν,α is a homeomorphism from Y onto Yν,α. More-over, Wν,αAν,αW−1ν,α = Aγ where(Aγf)(x) = −xf(x) + 2γ∫ ∞xf(y) dy ,A Fragmentation Equation 5with γ = 2(α+ 1)/(ν + 2) and D(Aγ) = {f ∈ Yν,α : xf ∈ Yν,α } .Lemma 1. If α > −1 then the operator(Aγ , D(Aγ)) = (Wν,αAν,αW−1ν,α,Wν,α[D(Aν,α])generates a stochastic semigroup {Sγ(t)}t≥0 = {Wν,αSν,α(t)W−1ν,α}t≥0 onYν,α. Moreover,Sγ(t)f = limn→∞(I − Pnα+1 + exp(Aγt)Pnα+1)f , f ∈ Yν,α , (8)where Pnα+1 is defined by (7).Proof. This follows from Theorem 3, Corollary 1 and formulae (5)–(7).An explicit formula for Sγ(t) can now be obtained using (8) and thepower series definition of exp(Aγt) . First we require a formula for positiveiterates of the operator Aγ .Lemma 2. For each k = 1, 2, 3, . . .,(Akγf)(x) = (−x)kf(x)+2(−1)k−1γk∑j=1(kj)(1− 2γ)j−1 xk−j∫ ∞x(y − x)j−1(j − 1)! f(y) dy,where (c)k = c(c+ 1) . . . (c+ k − 1) .Proof. The proof follows by induction.Lemma 3. For each t > 0 and f ∈ Yν,α,(Sγ(t)f)(x) = e−xt(f(x) +2tγ∫ ∞x1F1(1− 2γ, 2; t(x− y))f(y) dy),where 1F1 denotes the confluent hypergeometric function.Proof. This can be deduced from (8) and Lemma 2.Theorem 4. For each α > −1, ν ∈ (−2, 0], t > 0 and f ∈ Y ,(Sν,α(t)f)(x) = e−txα+1[f(x)+(ν + 2)t∫ ∞x1F1(α− ν − 1α+ 1, 2; t(xα+1 − yα+1))(xy)νyαf(y) dy ] .Moreover, the domain D(Aν,α) is positively invariant under the semigroup{Sν,α(t)}t≥0 .Proof. The formula given for Sν,α(t) follows from Lemma 3 and the factthat Sν,α(t) =W−1ν,αSγ(t)Wν,α . The invariance property can be establishedby a direct calculation.6 P.N. Blair, W. Lamb, I.W. StewartThe invariance of D(Aν,α) leads immediately to the following more sat-isfactory version of Theorem 2.Corollary 2. If α > −1 and ν ∈ (−2, 0] then the ACP (4) has theunique, non-negative mass-conserving solution c(t) = Sν,α(t)c0 for eachnon-negative c0 in D(Aν,α).References1. E.D. McGrady and R.M. Ziff, “Shattering” transition in fragmentation,Phys. Rev. Lett. 58 (1987), 892–895.2. K-J. Engel and R. Nagel, One-parameter semigroups for linear evolutionequations, Springer-Verlag, New York, 1999.3. W. Lamb and A.C. McBride, On a continuous coagulation and fragmen-tation equation with a singular fragmentation kernel, to appear in Recentcontributions to evolution equations, Marcel Dekker Lecture Notes Ser.4. D.J. McLaughlin, W. Lamb and A.C. McBride, A semigroup approachto fragmentation models, SIAM J. Math. Anal. 28 (1997), 1158–1172.5. J. Banasiak, On an extension of the Kato-Voigt perturbation theoremfor substochastic semigroups and its application, Taiwanese J.Math.5(2001), 169–191.

A semigroup approach to fragmentation models,

On a continuous coagulation and fragmentation equation with a singular fragmentation kernel, to appear in Recent contributions to evolution equations, Marcel Dekker Lecture Notes Ser.

An application of semigroup theory to a fragmentation equation

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