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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
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
Theprocessesconsideredbythecoagulationandfragmentationtheory
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
ApplicationstoCoagulation-fragmentation
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
Morecomplexapplications
Reddy, Banerjee, 2017
McKinley et al, 2016
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
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.
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
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)
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
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.
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
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
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
Somemathematicalwell-posednessresultsII
case: hypothesisandsetting: results: references:
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
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
Somemathematicalwell-posednessresultsIII
case: hypothesisandsetting: results: references:
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
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
Smoluchowskicoagulationequationandself-similarity
A relevantfeatureisthe roleoftheself-similarsolutions.
Thelongtermregimeinknownmodelsapproachestoaself-similarprofile.
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
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.
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
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
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
Theresultsoncoagulation
A visualoverview
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
Picture: self-similarityexponents
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
Picture: gellingsolutionsprofiles
Thesolutionsaremultipliedtimes ξ3−2ε− 1β . Twobehaviors: (I) polynomial
expansionattheorigin, (II) quasi-exponentialdecayatinfinity.
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
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.
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
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.
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
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.
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
SmallperturbationoftheproductKernel
Startingfromanexplicitsolution, welookforabranchofsolutionsperturbatively.
β (ε) = 2 + ελ (ε)
Φ (η) = Φ0 (η) + εΦ1 (η)
where Φ1 shouldbecontrolledinasuitablenormandthedecaypropertiesof Φ permitstudingthedecaypropertiesof ψ
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
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.
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
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.
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
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.
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
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|).
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
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.
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
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
<∞.
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
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.
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
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.
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
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
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
Weconsidernowthe purefragmentationequation.
Recall...
→֒
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
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.
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
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)
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
A generalwell-posednessresult
Both existence, uniqueness and convergence to the self-similarprofilescanbeobtainedundergeneralconditionsasithasbeendone in thework of Escobedo, Mischler andRodriguezRicard(2005).
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
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.
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
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+.
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
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
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
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
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
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).
Introduction IIºkindself-similarity: coagulation Iºkindself-similarity: fragmentation Extracontents
Conclusions: fragmentationequation
WeemployedWiener-Hopf’stechniquesinthecomplexplanetoachieve:
• explicitformulas;
• controlonasymptoticbehaviorsoftheself-similarsolution;
• regularityproperties(underthehypothesisofcontinuousfragmentationkernel);
• newunexpectedasymptoticsforthecompactsupportcase(aswellasadifferentexplicitformulafortheMellintransform).
Stillmanyinterestingproblemstostudy! Animportantone: whataboutnon-continuouskernels?