Selfsimilar solutions to coagulation and fragmentation...

Marco A. Fontelos

Selfsimilar solutions to coagulation and

fragmentation equations

Institute for Mathematics

Cons. Sup. de Investigaciones Científicas

Joint with G. Breschi, ICMAT, Madrid

Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents

Contents

IntroductionTheevolutionproblemsSelf-similarity

IIºkindself-similarity: coagulationA visualoverviewoftheresultsThegellingcaseNon-gellingcase

Iºkindself-similarity: fragmentationThelinearproblemPropertiesofthesolutionsThecompactsupportcase


Theprocessesconsideredbythecoagulationandfragmentationtheory


ApplicationstoCoagulation-fragmentation


Morecomplexapplications

Reddy, Banerjee, 2017

McKinley et al, 2016


Thecoagulation-fragmentationequationOnecanwritedownacoagulation-fragmentationequationofbalanceofmass:

d

dtc (x, t) = QC (c)−QF (c) ,

QC (c) :=1

2

x∫

0

K (x− y, y) c (x− y, t) c (y, t) dy − c (x, t)

∞∫

0

K (x, y) c (y, t) dy

QF (c) := β (x) c (x, t)−

∞∫

x

c (y, t)β (y)

yB

(

x

y

)

dy

where β (x) isthefragmentationrateofclustersofsize x and B (u) isthedaughterfragmentsdistributionkernel.Themassdependingandsymmetricoperator K(x, y) isknownasthecoagulationkernelanddescribestherateatwhichparticlesofthegivensizecoagulate.

Unfortunately, mostofthephysicallyinterestingmodelscorrespondtoequationsthatarenotexactlysolvable.


Purecoagulation: theSmoluchowskiequation

• Ithasbeenpossibletodiscoveranalyticalsolutionsexclusivelyforalimitednumberofkernels; theprincipalcasesinclude:Kc(x, y) = 1, K+ (x, y) = x+ y, K× (x, y) = xy.

• Formoregeneralkernels, studieshaveessentiallyreliedonnumericalmethods.

• Existence, positivityanduniquenessresultforalltimeforkernelsgrowingatmost linearly.

interests:

• singularities infinitetimeoccurforsomeKernels: verystrongcoagulationcreatesinfinitelydenseclusters.

• self-similarsolutions.

Laplace transform:

Regularized Laplace transform:

Fractional Burgers equation:

Nonlocal transport equation: MAF, Eggers, to appear in Nonlinearity 2019

Existence of singularities: A. Córdoba,

D. Córdoba, MAF, Ann. of Math. 2005

(Toy model for Euler eqn.)

Selfsimilarity of the second Kind (Barenblatt)


Singularity=gelation

gelation:

Starting from an initial particle size distribution c0 = c (x, 0) suchthatallitsmoments Mi(0) =

∫

∞

0xic0 (x) dx arebounded, thereisa

certaintime t∗ suchthatallmoments Mi(t) for i ≥ i0 diverge.

Thedistribution c (x, t) developsan heavyalgebraictailinfinitetime.

Upto t < t∗ totalconservationofmassisguaranteed, butafter t∗ thefirstmoment M1 isnolongerconservedandstartstodecrease.

Thisphenomenoniscalledfinitetimegelationandindicatestheaggregationoftheparticlesinaclusterofinfinitedensitythatdrainsmassfromthecoagulatingsystemwithfinite x-mass.

Aninterestingproblemisthecontinuationof post-gellingsolutions.


Somemathematicalwell-posednessresultsI

case: hypothesisandsetting: results: references:

purecoagulatio

n

asymptoticallysub-linearvsasymptoticallysub-quadratic:

K(x,y)≤κ0(1+x+y),or κ0(1+x+y)<K(x,y)≤κ1xy

∃!, T ∗=∞

∃!, T ∗<∞

(estimation)

Classicreference:e.g. Norris99

homogeneousnon-gellingcoagulation:

K(x,y)≤κ0(xλ+yλ), λ∈(−∞,1]∃! weaksense,T ∗=∞

Fournier-Laurençot2006

homogeneousgellingcoagulation:

κ1(xy)λ2 ≤K(x,y)≤κ2(xyλ−1+xλ−1y),

with λ thedegreeofhomogeneityof Kand λ∈(1,2]

∃! weaksense,T ∗<∞,lowerbound

Fournier-Laurençot2006

faster-growing,singularkernel:

variousformulations, forexample:K(x,y)≥A((1+x)λ+(1+y)λ), λ>1

instantaneousgellingsolutions

CarranddaCosta1992


Somemathematicalwell-posednessresultsII


purefragmentatio

n

singularrateoffragmentation:

β(x)=xλ, λ<0

”shattering”critictimeisexplicitlyknown

McGrady–Ziff87Cheng–Redner90

non-singularcase:massconservingdistribution∫ x0 yb(y,x) dy=x

well-posedin L1xdx

mass-conservationLamb2004

homogeneousratewithinfinitefragments:

ν∈(−2,−1] and B(u)=(ν+2)uν well-posedness Cepeda2014

finitepositivemomentsinthenon-singularcase:

Mk(c0)<∞ forall k∈N Mk<∞

EquicontinuoussemigroupsinFréchetspaces:Banasiak-Lamb2014


Somemathematicalwell-posednessresultsIII


coagulation-fragmentatio

n

fragmentationdominatesovercoagulation:

K(x,y)≤κ1(xy)λ,β(x)≥κ2x

γ,λ∈[0,1],γ∈(−1,∞),2+γ>2λ

amassconservingsolutionexists:T ∗=∞

Manystrategies,EscobedoLaurençotandPerthame2003

strongfragmentationandweakcoagulation:

β(x)=xλ,B(u)=(ν+2)uν ,ν∈(−1,0],0≤σ≤1,σ<λ, andK(x,y)≤κ2((1+x)σ+(1+y)σ)

global-in-timeandunicity

Banasiak, LambandLanger2013

strongfragmentationandstrongcoagulation:

thesameasaboveand0≤σ≤τ<λ andK(x,y)≤κ1(xσyτ+xτyσ)

localsolutionanduniqueness

asabove

detailedbalancecondition:

let M∈L11beanon-zerofunctionK(x,y)M(x)M(y)=b(x,y)M(x+y).

solutionsconvergetoanequilibrium:c[M1(0)] (x)

dependingonlyontheinitialmass

Carr-daCosta94,Dubowski-Stewart96

otherequilibria:strongfragmentation, smallinitialmass(thelatterisatechnicalcondition)

solutionsconvergetoanequilibriumc[M1(0)] (x)

L2 exponentialtrend

Fournier–Mischler2004


Smoluchowskicoagulationequationandself-similarity

A relevantfeatureisthe roleoftheself-similarsolutions.

Thelongtermregimeinknownmodelsapproachestoaself-similarprofile.


Self-similarapproach: gellingcase

• Homogeneusmultiplicativekernel K (x, y) = (xy)1−ε; for 0 ≤ ε < 1

2

weareingelationregime. Possiblesolutionscanbehaveasymptoticallyas t→ t∗:

c (x, t) = (t∗ − t)αψ(

(t∗ − t)βx)

• Inthiscase, theself-similarsolutionshavenotbeendeterminedexplicitlyand, moreremarkably, fromnumericalexperiments, thesimilarityexponentscannotbedeterminedfromsimpledimensionalconsiderations(cf. Lee, 2001).

• Sotheyseemtobelongtotheclassofthesocalled self-similarityofthesecondkind.

• Remember: when ε < 0, thestrongcoagulationrateyields instantaneous

gelation.


Substituting:

−(β(3−2ε)−1)ψ(ξ)−βξψ′(ξ)=

= 12

∫[0,ξ](ξ−y)

1−εψ(ξ−y) y1−εψ(y) dy−ξ1−εψ(ξ)∫R+ y1−εψ(y) dy.

where, again, α = (3− 2ε)β − 1, fromdimensionalconsidera-tions.

Wehave β < 0 innon-gellingcasesand β > 0 otherwise. Therefore,westudytheself-similarproblemforall λ−multiplicativekernels:

|β| T [ψ] + QC,λ [ψ] = 0


Theresultsoncoagulation

A visualoverview


Picture: self-similarityexponents


Picture: gellingsolutionsprofiles

Thesolutionsaremultipliedtimes ξ3−2ε− 1β . Twobehaviors: (I) polynomial

expansionattheorigin, (II) quasi-exponentialdecayatinfinity.


Thegellingcaseresult: mainingredients

Recall:

− (β (3− 2ε)− 1)ψ(ξ)− βξψ′(ξ) =

=1

2

∫

[0,ξ]

(ξ − y)1−ε ψ (ξ − y) y1−εψ (y) dy−ξ1−εψ (ξ)

∫

R+

y1−εψ (y) dy.


Laplacetransform

Westudiedthes.-s. solutionstoSmoluchowskiequationclosetotheproductkernel. Ourstrategyinvolvesa regularizedLaplacetransform:

• Transform: Φ(η) = −∫

R+

(e−ηξ − 1)ξψ (ξ) dξ.

• Inversetransform: ψ (ξ) = 12πi

1ξ2

∫ i∞

−i∞eηξΦ′ (η) dη.

one formally derives a new problem:

− ((1− 2ε)β − 1)Φ (η) + βηΦ′ (η) =1

2

d

dη

[

D−εη Φ(η)

]2,

with:

D−εη Φ(η) = −

∫

R+

(e−ηξ − 1)ξ1−εψ (ξ) dξ.


TheexplicitcaseIf ε = 0, wecandeducepreviouslyknownresults:Laplacevariableequationreducestoafirstorderordinarydifferentialequation,andwecaneasilyobtaintheimplicitexpression:

η(Φ) = kΦβ

β−1 +Φ

Withthe specificelectionof β = 2, thisequationisasecondorderpoly-nomial; wecanfinditszerosandobtaintwopossiblesolutions, butweonlyconsidertheonegivingapositive ψ:

Φ0 (η) = 2π

(

−1 +

√

1 +η

π

)

, ψ (ξ) =1

ξ52

e−πξ.

Wecanalsostudytheequationfordifferentvaluesof β, buttheinversetransform ψ presentsalgebraictails.


SmallperturbationoftheproductKernel

Startingfromanexplicitsolution, welookforabranchofsolutionsperturbatively.

β (ε) = 2 + ελ (ε)

Φ (η) = Φ0 (η) + εΦ1 (η)

where Φ1 shouldbecontrolledinasuitablenormandthedecaypropertiesof Φ permitstudingthedecaypropertiesof ψ


ThefirstfixedpointtheoremTheorem. B.andFontelos

Thereexistsan ε0 > 0 andafunction

λ(ε) = 2 +O(ε)

suchthatforany 0 < ε < ε0 andwith β(ε) = 2 + ελ(ε) thereexistsauniquesolutiontoSmoluchowski’sself-similarequation(uptorescaling)satisfying:

∞∫

0

ξ72− δ

2ψ (ξ) dξ <∞

forany 0 < δ ≪ 1.

Boththe behaviourof Φ for η ∼ 0 andatinfinitymustbecontrolled. Then, decayingpropertiesof Φ giveestimatesonmomentslikethe 7

2-thabove.


Settingofthenonlinearproblem

Let Φ(η) = Φ0 (η) + εΦ1 (η). WegettheLaplaceself-similarequationfor Φ1:

Φ1 − 2ηd

dηΦ1 +

d

dη(Φ0Φ1) = F0(η) + εLΦ1 + εQ(Φ1,Φ1)

Linearandquadratictermsin Φ1 are O (ε), so, calling N [Φ1] therighthandside, wedefineamappingthatassignstoagiven Φ1 thesolutionto

Φ1 − 2ηd

dηΦ1 +

d

dη(Φ0Φ1) = N

[

Φ1

]

thisisdoneinthefollowing.


Steps

• EstablishaHardy-likeinequality:Mγ ≤

∫∞

−∞(W (|k|))2 |Φ′ (ik)|2 dk;

• thisgivesafunctionalspace Y tolookforasolution Φ (η).

• Thesolutionmustbeclose(asphereofsmallradius)toasuitablefunction h (η);

• anditsbehavior closetozero determinesthedecayof ψ (ξ) atinfinity. Wewant ψ todecayfasterthansomepower.

• Ananalyticityconditionon h (η) -thatis: eliminatingan η2 log ηterm-fixesthevalueof β andmakes ψ decayfasterthansuchpower.

• Nowuseafixed-pointtheorem(thetrickypartistodealwith h atthesametime).

• Finally, usethemoment-equationtoobtainthestrongresult: allhighermomentsarealsobounded.


Analysisoftheexpansion:

1. Given Φ1(η) = a1η − 1πa1η

2 + o(|η|2) onecancompute(

NΦ1

)

(η) = b1η + b2η2 +O(|η|3) as η → 0;

2. Consider ρ(η) = b1η+b2η2

1− η2

2

and

h(η) ≡ (√

1+ ηπ−1)

2

√1+ η

π

∫ η

η0

√1+w

π

(√

1+wπ−1)

2

ρ(w)Φ0(w)−2wdw;

3. Then, integratingthedifferentialequationfor Φ1,

Φ1 (η)−h(η) =(√

1 + ηπ− 1)2

√

1 + ηπ

∫ η

0

√

1 + wπ

(√

1 + wπ− 1)2

(

NΦ1

)

(w)− ρ(w)

Φ0 (w)− 2wdw;

4. Then Φ1 − h isa o(

η2)

andwecancomputeexplicitly

h(η) = b1η −1

πb1η

2 −(

b2 +3

4πb1

)

η2 ln η +O(|η|3 log |η|).


TheanalyticityconditionThepresenceofthe η2 ln η logarithmictermintheLaplacesolutionimpliesthatthecorrespondinginversetransformpresentsanalgebraicdecay:

ψ (ξ) ≃ξ→∞

sin (|ε|π)ξ4+O(ε)

.

Fora fasterdecay,

b2 +3

4πb1 = 0

whichfixes β (ε) = 2 + 2ε+O(

ε2)

. Higherorderscanbealso explicitlycomputed imposingmoretermsintheexpansionof Φ1. This analyticitycondition characterizestheself-similarproblemasoneofthe secondkind.


ThenormWeobtainestimatesof (Φ1 (η)− h (η)) intermsofJ(η) ≡

(

N[

Φ1

]

(η)− ρ(η))

inanappropriatefunctionalspace:

DefinitionLet X bethespaceoffunctions f suchthat:

∥f(k)∥X ≡∞∫

−∞

(

1

|k|3+

1

|k|32

)2

|f(k)|2 dk <∞.

Let Y bethesubspaceof X whosefunctionsaresuchthat:

∥f (k)∥Y ≡ ∥f (k)∥X +

∥

∥

∥

∥

kd

dkf (k)

∥

∥

∥

∥

X

<∞.


Howtoobtainthe 72 − th moment

ByBanach’sfixed-pointtheorem, wefindthesolutiontothenonlinearproblemsuchthat ∥Φ1 (ik)− h (ik)∥Y <∞ andhence:

∞∫

−∞

(

1

|k|2+

1

|k|12

)2∣

∣Φ′1 (ik)− h′ (ik)

∣

∣

2dk <∞.

Toconcludethetheorem, weemploy

∫

R+

ξ(γ+2) |ψ (ξ)| dξ < C

∫

R

∣

∣Φ′ (k)∣

∣

2(

|k|1−2γ−δ + |k|1−2γ+δ)

dk

with γ = 32 − δ

2 and 0 < δ ≪ 1.


Fasterdecayfortheself-similarsolution

Theorem. B.andFontelos

Undertheprevioushipotesis, wecanconcludethatforany |ε| <ε0 andwith the same β(ε) = 2 + ελ(ε) foundbefore, thereexistsauniquesolution(uptorescaling)withallitsmoments Mα

(α ≥ 2)bounded.

Remarkably, nofurtherrestriction isplacedupon ε0 or β.


Wemultiplythes.s. Smoluchowskiequationatbothsidesby ξα with α > 52

andintegratetoobtain:

(

α− 2 + 2ε+1

β

)

Mα =1

2β

∫

R+

∫

[0,ξ]

((ξ − y) + y)α (ξ − y)1−ε y1−εψ (ξ − y)ψ (y) dydξ

−1

βMα+1−εM1−ε

andweuse:

(ξ − y)α + yα + C1

(

(ξ − y)α−1 y + yα−1 (ξ − y))

≤ ((ξ − y) + y)α ≤

≤ (ξ − y)α + yα + C2

(

(ξ − y)α−1 y + yα−1 (ξ − y))

.

Usingtheprecedinginequalities, wecanbound:

C1

βMα−εM2−ε ≤

(

α− (−2ε+ 2) +1

β

)

Mα ≤C2

βMα−εM2−ε


Weconsidernowthe purefragmentationequation.

Recall...

→֒


Theoriginalequation:

d

dtc (x, t) = −β (x) c (x, t) +

∞∫

x

c (y, t)β (y)

yB

(

x

y

)

dy

with β (x) = γxγ , and γ ≥ 0 (non-shatteringcase).B : [0, 1] → R

+ isthe relativefragmentationrate andverifiesthenormalizationproperty:

1∫

0

uB (u) du = 1.


Gettingtheself-similarfragmentationequationWeseekafunctionlike (t′)α φ

(

x

(t′)β

)

suchthatthemassmoment M1, andalsoallthe

othermomentsareconserved.

Suchsymmetryrequirementpermitsdeterminingtheself-similarformulationsothatthe

similarityproblemdoesactuallybelongtothe firstkindself-similarity.

a self-similar problem of the first kind:

c (x, t) = t2

γφ (ξ) ,

with ξ = xt1

γ . Imposingintheevolutionequation, weget:

2φ (ξ) + ξd

dξφ (ξ) = −γξγφ (ξ) +

∞∫

ξ

φ (η) γηγ−1 B

(

ξ

η

)

dη

thatcanbealsowritten:

T [φ] (x) = QF [φ] (x)


A generalwell-posednessresult

Both existence, uniqueness and convergence to the self-similarprofilescanbeobtainedundergeneralconditionsasithasbeendone in thework of Escobedo, Mischler andRodriguezRicard(2005).


TheMellintransformapproach

• Considerthetransform Φ(z) =∫

∞

0xz−1φ (x) dx.

• Wealsodefine Θ(z) =∫

1

0uz−1B (u) du.

Thenewproblem

(2− z) Φ (z) = (Θ (z)− 1) γΦ(z + γ) ,

Weareinterestedinevaluating Φ at z = is inordertoapplytheinverseMellintransform:

φ (x) =1

2πi

∫ i∞

−i∞

x−zΦ(z) dz.


TheCarlemanproblemSomereferencestoWiener-Hopfmethodsandhowtosolvethisclassofproblemscan

befoundinEscobedoVelázquez(2010).

Firstofall, weintroducethenewvariable ζ = ez·2πi· 1

γ anddefinef (ζ) = Log (Φ (z)), T (ζ) = Θ (z).TheCarlemanproblemnowreads:

f (ζ + 0·i)− f (ζ − 0·i) = Log

(

γT (ζ)− 1

2− γ2πi

Logζ

)

= Logγ + Log (T (ζ)− 1)− Log(

2−γ

2πiLogζ

)

,

where f isanalyticin C \ R+.


Thetypicalstrategyandavariation

Thisproblembelongstothegeneralclassofproblemsoftheform:

f (ζ + 0·i)− f (ζ − 0·i) = G(ζ)

withsolution

f (ζ) =1

2πi

∞∫

0

G(s)ds

s− ζ,

provided G(s) decayssufficientlyfastatinfinitysothattheintegraliswelldefined.

0 1

b(u)

Mitosis

z0

.

.

.

.

.

.

.

.

.

.

0 1

b(u)

u0


Propertiesoftheself-similarsolution φ

Theself-similarsolution φ (ξ) givenastheinverseMellintransformof Φverifies:

∞∫

0

eξγ φ (ξ) dξ

ξ1+µ−τ + ξ1+µ+τ< ∞

forany 0 < τ < min {µ, γ}.Moreover, Mα =

∫∞0 ξαφ (ξ) dξ < ∞ foreach α > −µ − 1. The

solutionhasthefollowing regularityproperty: forall l > −µ+ n,

dn

dξn

(

ξlφ (ξ))

≤ 1

2π

∫∞

−∞|(n−is)(n−1−is)···(1−is)||Φ(is+l−n)|ds < ∞,

oralso∣

∣

∣

∣

dn

dξnφ (ξ)

∣

∣

∣

∣

≤ Cnξµ−n−δ, with δ > 0, δ ≪ 1.

Reddy, Banerjee, 2017

McKinley et al, 2016


Conclusions: Smoluchowskiequation

Weproposea generalscalinghypothesis forcoagulation:

• A firstexistenceanduniquenesstheoremforrapidlydecayingself-similarsolutionsinthegellingrange.

• Asymptoticandanalyticalresultsinwiderangesofhomogeneity λdegree.

• Differenttechniquestocompute β.

• Numericalevidences.

Ourworksonthissubject:

• Breschi-Fontelos, Nonlinearity2014. Self-similarsolutionsofthesecondkindrepresentinggelationinfinitetimefortheSmoluchowskiequation.

• Breschi-Fontelos. Onglobalintimeself-similarsolutionsofSmoluchowskiequationwithmultiplicativekernel. (inPhD theses, tosubmit).


Conclusions: fragmentationequation

WeemployedWiener-Hopf’stechniquesinthecomplexplanetoachieve:

• explicitformulas;

• controlonasymptoticbehaviorsoftheself-similarsolution;

• regularityproperties(underthehypothesisofcontinuousfragmentationkernel);

• newunexpectedasymptoticsforthecompactsupportcase(aswellasadifferentexplicitformulafortheMellintransform).

Stillmanyinterestingproblemstostudy! Animportantone: whataboutnon-continuouskernels?

Selfsimilar solutions to coagulation and fragmentation...

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