Weak compactness techniques for
coagulation equations.
II. The continuous coagulation equation
Philippe Laurençot
Institut de Mathématiques de Toulouse
26.07.2013
(IMT)
26.07.2013
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The continuous coagulation equation
Discrete coagulation equation i ∈ N, i ≥ 1:
i−1
∞
j=1
j=1
X
dfi
1X
K (j, i − j) fj fi−j −
K (i, j) fi fj .
=
dt
2
Continuous coagulation equation x ∈ (0, ∞):
Z
1 x
∂t f (t, x) =
K (y , x − y ) f (t, y ) f (t, x − y ) dy
2 0
Z ∞
−
K (x, y ) f (t, x) f (t, y ) dy .
0
−→ integro-partial differential equation
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The continuous coagulation equation
∂t f (t, x) =
1
2
x
Z
K (y , x − y ) f (t, y ) f (t, x − y ) dy
Z0
−
∞
K (x, y ) f (t, x) f (t, y ) dy .
0
Coagulation kernel: K (x, y ) = K (y , x) ≥ 0
Total number of clusters M0 (f ) and total mass M1 (f ):
Z ∞
Z ∞
M0 (f ) =
f (y ) dy ,
M1 (f ) =
yf (y ) dy .
0
(IMT)
0
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Coagulation kernel K : examples
K (x, y ) =
x
1/3
+y
1/3
1
1
+
x 1/3 y 1/3
(Smoluchowski, 1916)
K (x, y ) = 2
K (x, y ) = (ax + b)(ay + b) , a > 0 , b ≥ 0
3
K (x, y ) =
x 1/3 + y 1/3
2 1/3
1/3 K (x, y ) =
x 1/3 + y 1/3
x
−
y
K (x, y ) = x α y β + x β y α ,
(Stockmayer, 1943)
α ≤ 1,β ≤ 1.
Unboundedness as (x, y ) → ∞ or (x, y ) → (0, 0).
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Solvable kernels
K (x, y ) = 2. “Desingularized” Laplace transform.
Z ∞
L(t, s) :=
1 − e−sx f (t, x) dx −→ ∂t L = −L2 .
0
K (x, y ) = x + y . “Desingularized” Laplace transform.
Z ∞
L(t, s) :=
1 − e−sx f (t, x) dx −→ ∂t L = L∂s L − L .
0
K (x, y ) = xy . “Desingularized” Laplace transform.
Z ∞
Λ(t, s) :=
1 − e−sx xf (t, x) dx −→ ∂t Λ = Λ∂s Λ .
0
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Outline
1
Existence: Bounded coagulation kernels
2
Existence: Unbounded coagulation kernels
3
Weak compactness in Lp , p ∈ [1, ∞]
4
Back to the discrete coagulation equation
5
Existence: Unbounded coagulation kernels
(IMT)
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Existence: Bounded coagulation kernels
Outline
1
Existence: Bounded coagulation kernels
2
Existence: Unbounded coagulation kernels
3
Weak compactness in Lp , p ∈ [1, ∞]
4
Back to the discrete coagulation equation
5
Existence: Unbounded coagulation kernels
(IMT)
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Existence: Bounded coagulation kernels
Existence
Differential equation in an infinite dimensional Banach space
Theorem
Let (E, k.k) be a Banach space, G : E → E a locally Lipschitz
continuous function, and f in ∈ E. There is a unique maximal solution
f ∈ C 1 ([0, Tm ); E) to the differential equation
df
(t) = G(f (t)) ,
dt
t ∈ [0, Tm ),
f (0) = f in ,
and:
Tm = ∞
OR
Tm < ∞ and
lim kf (t)k = ∞ .
t→Tm
Consequence of the contraction mapping principle.
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Existence: Bounded coagulation kernels
Existence
df
= P(f ) − f L(f ) , f (0) = f in ,
dt
with P(f ) and L(f ) given by
Z
1 x
P(f )(x) =
K (y , x − y ) f (y ) f (x − y ) dy ,
2
Z ∞0
L(f )(x) =
K (x, y ) f (y ) dy ,
x >0.
x >0,
0
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Existence: Bounded coagulation kernels
Existence
For µ ∈ R,
L1µ (0, ∞)
:=
Z
1
f ∈ L (0, ∞) : kf k1,µ =
∞
(1 + x )|f (x)| < ∞ .
µ
0
If
K (x, y ) ≤ κ ,
x > 0 , y > 0,
P and f 7→ f L(f ) are locally Lipschitz continuous from L10 (0, ∞) to
L10 (0, ∞).
f 7→ P(f )+ is also locally Lipschitz continuous from L10 (0, ∞) to
L10 (0, ∞), where r+ := max {r , 0} denotes the positive part of a
real number r .
Z
∞
Z
∞Z ∞
[P(f )(x) − f (x) L(f )(x)] dx = −
0
K (x, y )f (x)f (y ) dydx .
0
(IMT)
0
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Existence: Bounded coagulation kernels
Existence
Theorem
If
K (x, y ) ≤ κ ,
x > 0 , y > 0,
and
f in ∈ L10 (0, ∞) ,
f in ≥ 0 a.e. in (0, ∞) ,
there is a unique global solution f ∈ C 1 ([0, ∞); L10 (0, ∞)) such that
f (t, x) ≥ 0 a.e. in (0, ∞) for all t ≥ 0. Furthermore, if f in ∈ L11 (0, ∞),
then
f (t) ∈ L11 (0, ∞) and M1 (f (t)) = M1 (f in ) , t ≥ 0 .
Here, for f ∈ L1µ (0, ∞),
Z
Mµ (f ) :=
∞
x µ f (x) dx .
0
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Existence: Bounded coagulation kernels
Existence: proof
Apply the existence theorem with
G(f ) := P(f )+ − f L(f ) ,
Non-negativity follows and then implies P(f )+ = P(f ),
Non-negativity −→ global existence.
Conservation of mass −→ Fubini theorem.
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Existence: Bounded coagulation kernels
Further properties
Easy consequence of Fubini’s theorem:
Lemma
If ϑ ∈ L∞ (0, ∞) then
Z ∞
d
ϑ(x)f (t, x) dx
dt 0
Z Z
1 ∞ ∞
=
[ϑ(x + y ) − ϑ(x) − ϑ(y )] K (x, y )f (t, x)f (t, y ) dydx
2 0
0
−→ t 7→ M0 (f (t)) is non-increasing.
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Existence: Bounded coagulation kernels
Further properties
Other formal consequences of the previous lemma are:
If µ ∈ (−∞, 1), then
t 7−→ Mµ (f (t)) is non-increasing.
If µ ∈ (1, ∞), then
t 7−→ Mµ (f (t)) is non-decreasing.
If α > 0,
Z
t 7−→
∞
(eαx − 1) f (t, x) dx is non-decreasing.
0
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Existence: Bounded coagulation kernels
Decay of the total number of clusters
Assume further that, for each η > 0, there is δη > 0 such that
K (x, y ) ≥ δη > 0 ,
x >η, y >η.
Then
lim M0 (f (t)) = 0 .
t→∞
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Existence: Unbounded coagulation kernels
Outline
1
Existence: Bounded coagulation kernels
2
Existence: Unbounded coagulation kernels
3
Weak compactness in Lp , p ∈ [1, ∞]
4
Back to the discrete coagulation equation
5
Existence: Unbounded coagulation kernels
(IMT)
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Existence: Unbounded coagulation kernels
Unbounded coagulation kernels
The previous approach again does not seem to extend to unbounded
coagulation kernels −→ compactness method.
1
Build a sequence of approximations of the original problem:
which depends on a parameter N ≥ 1,
for which the existence of a solution is “easy” to show,
and “converges” to the original problem as N → ∞.
2
Derive estimates which are independent of N ≥ 1 and guarantee
compactness with respect to the size variable x and the time
variable t.
3
Show convergence as N → ∞.
Remark. L10 (0, ∞) 6⊂ L∞ (0, ∞)
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Weak compactness in Lp , p ∈ [1, ∞]
Outline
1
Existence: Bounded coagulation kernels
2
Existence: Unbounded coagulation kernels
3
Weak compactness in Lp , p ∈ [1, ∞]
4
Back to the discrete coagulation equation
5
Existence: Unbounded coagulation kernels
(IMT)
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Weak compactness in Lp , p ∈ [1, ∞]
Weak convergence
Definition
Let p ∈ [1, ∞). A sequence (fn )n≥1 in Lp (0, ∞) converges weakly
to f (fn * f ) in Lp (0, ∞) if
Z ∞
Z ∞
fn (x) ϑ(x) dx =
f (x) ϑ(x) dx
lim
n→∞ 0
0
for all ϑ ∈ Lp/(p−1) (0, ∞).
?
A sequence (fn )n≥1 in L∞ (0, ∞) converges ?-weakly to f (fn * f )
in L∞ (0, ∞) if
Z ∞
Z ∞
lim
fn (x) ϑ(x) dx =
f (x) ϑ(x) dx
n→∞ 0
0
for all ϑ ∈ L1 (0, ∞).
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Weak compactness in Lp , p ∈ [1, ∞]
Weak compactness: p ∈ (1, ∞)
Finite dimensional spaces 6= infinite dimensional spaces.
Theorem
Let p ∈ (1, ∞) and consider a bounded sequence (fn )n≥1 in Lp (0, ∞).
Then there is a subsequence of (fn )n≥1 which converges weakly in
Lp (0, ∞).
This follows from Kakutani’s theorem as Lp (0, ∞) is reflexive.
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Weak compactness in Lp , p ∈ [1, ∞]
Weak compactness: p = ∞
Theorem
Consider a bounded sequence (fn )n≥1 in L∞ (0, ∞). Then there is a
subsequence of (fn )n≥1 which converges ?-weakly in L∞ (0, ∞).
This follows from the Banach-Alaoglu-Bourbaki theorem as L∞ (0, ∞)
is the dual of the separable space L1 (0, ∞).
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Weak compactness in Lp , p ∈ [1, ∞]
Weak compactness: p = 1
Concentration. Consider ϕ ∈ C0∞ (0, ∞) such that ϕ ≥ 0 and
kϕk1,0 = 1. For n ≥ 1 and x > 0, define ϕn (x) = nϕ(nx). Then
kϕn k1,0 = 1 and
lim ϕn (x) = 0 ,
n→∞
x >0.
If ϕn * ψ in L1 (0, ∞) then ψ ≥ 0 a.e. in (0, ∞) and
Z ∞
Z ∞
ψ(x) dx = lim
ϕn (x) dx = 1
n→∞ 0
0
Z ∞
Z ∞
ψ(x) dx = lim
ϕ(x) dx = 0 ,
n→∞ nr
r
r >0,
and a contradiction. In fact,
ϕn * δx=0 in the sense of distributions.
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Weak compactness in Lp , p ∈ [1, ∞]
Weak compactness: p = 1
Vanishing. For n ≥ 1 and x > 0, define ξn (x) = e−|x−n| . Then
kξn k1,0 = 2 − e−n and
lim ξn (x) = 0 ,
x >0.
If ξn * ψ in L1 (0, ∞) then ψ ≥ 0 a.e. in (0, ∞) and
Z ∞
Z ∞
ψ(x) dx = lim
ξn (x) dx = 2
n→∞ 0
Z0 r
Z r
ψ(x) dx = lim
ex−n dx = 0 ,
r >0,
0
n→∞
n→∞ 0
and a contradiction.
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Weak compactness in Lp , p ∈ [1, ∞]
Weak compactness: p = 1
Concentration and vanishing are somehow the only obstructions to
weak compactness in L1 .
Theorem (Eberlein-Šmulian)
Let E be a Banach space such that every bounded sequence has a
subsequence converging in the σ(E, E 0 )-topology. Then E is reflexive.
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Weak compactness in Lp , p ∈ [1, ∞]
Uniform integrability
Definition (Uniform integrability)
A bounded sequence (fn )n≥1 in L1 (0, ∞) is uniformly integrable in
L1 (0, ∞) if
Z
|fn (x)| dx = 0 .
lim sup
c→∞ n≥1
{|fn |≥c}
Concentration.
Z
Z
|ϕn (x)| dx =
{|ϕn |≥c}
(IMT)
|ϕ(x)| dx −→ 1 6= 0 .
{|ϕ|≥c/n}
n→∞
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Weak compactness in Lp , p ∈ [1, ∞]
Uniform integrability: examples
Let (fn )n≥1 be a bounded sequence in L1 (0, ∞).
If (fn )n≥1 is bounded in Lp (0, ∞) for some p > 1, then (fn )n≥1 is
uniformly integrable.
If there is F ∈ L1 (0, ∞) such that |fn | ≤ |F | a.e. for all n ≥ 1, then
(fn )n≥1 is uniformly integrable.
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Weak compactness in Lp , p ∈ [1, ∞]
Uniform integrability
Definition (Modulus of uniform integrability)
Let (fn )n≥1 be a bounded sequence in L1 (0, ∞). For ε > 0, we set
Z
|fn (x)| dx : n ≥ 1 , |A| ≤ ε ,
η{(fn ), ε} := sup
A
and we define the modulus of uniform integrability η{(fn )} of (fn )n≥1 by
η{(fn )} := lim η{(fn ), ε} = inf η{(fn ), ε} .
ε→0
ε>0
Proposition
A bounded sequence (fn )n≥1 in L1 (0, ∞) is uniformly integrable if and
only if η{(fn )} = 0.
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Weak compactness in Lp , p ∈ [1, ∞]
Weak compactness: p = 1
Theorem (Dunford-Pettis)
Let (fn )n≥1 be a bounded sequence in L1 (0, ∞). The following two
statements are equivalent:
(a) There is a subsequence of (fn )n≥1 which converges weakly in
L1 (0, ∞).
(b) The sequence (fn )n≥1 enjoys the following two properties:
η{(fn )} = 0
and, for every ε > 0, there is Rε > 0 such that
Z ∞
sup
|fn (x)| dx ≤ ε.
n≥1
(IMT)
Rε
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Weak compactness in Lp , p ∈ [1, ∞]
Consequences
Corollary
Let (gn )n≥1 and (hn )n≥1 be two sequences of measurable functions
such that
gn * g in L1 (0, ∞).
hn −→ h a.e. in (0, ∞) and |hn (x)| ≤ C for all n ≥ 1 and
x ∈ (0, ∞). Then
gn hn * gh in L1 (0, ∞) .
The proof combines Egorov’s theorem with the Dunford-Pettis theorem.
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Weak compactness in Lp , p ∈ [1, ∞]
Remarks
Theorem (Egorov)
Let R > 0 and (hn )n≥1 be a sequence of measurable functions in
(0, R) such that hn −→ h a.e. in (0, R). Then, for each ε > 0, there is a
measurable subset Aε of (0, R) such that |Aε | ≤ ε and
lim
sup
n→∞ x∈(0,R)\A
|hn (x) − h(x)| = 0 .
ε
The previous corollary is somehow an extension of the following result:
Let p ∈ (1, ∞). If gn * g in Lp (0, ∞) and hn −→ h in Lp/(p−1) (0, ∞),
then gn hn * gh in L1 (0, ∞).
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Weak compactness in Lp , p ∈ [1, ∞]
The de la Vallée-Poussin theorem
Theorem
Let (fn )n≥1 be a bounded sequence in L1 (0, ∞). The following two
statements are equivalent:
(fn )n≥1 is uniformly integrable.
There exists a convex function Φ ∈ C ∞ ([0, ∞)) such that
Φ(0) = Φ0 (0) = 0, Φ0 is a concave function,
Φ0 (r ) > 0 if r > 0 ,
Φ(r )
lim
= lim Φ0 (r ) = ∞ ,
r →∞ r
r →∞
Z
∞
Φ(|fn (x)|) dx < ∞ .
sup
n≥1
(IMT)
0
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Weak compactness in Lp , p ∈ [1, ∞]
Example
Let α ∈ (0, 1). Then
f (x) =
1(0,1) (x)
,
xα
x >0
belongs to L1 (0, 1) and so does Φ(f ) where Φ(r ) = r 1+δ and
0 < δ < min {1, (1 − α)/α}.
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Weak compactness in Lp , p ∈ [1, ∞]
Consequence
Proof. Look for Φ of the form
Φ0 (r ) = Am r + Bm ,
r ∈ [Nm , Nm+1 ] ,
m≥0.
Corollary
Let (yn )n≥1 ∈ `10 . There exists an increasing sequence of positive real
numbers (λn )n≥1 such that λn → ∞ and
∞
X
λn |yn | < ∞ .
n=1
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Weak compactness in Lp , p ∈ [1, ∞]
A class of convex functions
Definition
We define CVP as the set of convex functions Φ ∈ C ∞ ([0, ∞)) with
Φ(0) = Φ0 (0) = 0 and such that Φ0 is a concave function satisfying
Φ0 (r ) > 0 for r > 0.
Definition
The set CVP,∞ denotes the subset of functions in CVP satisfying the
additional property
Φ(r )
= lim Φ0 (r ) = ∞
r →∞ r
r →∞
lim
Examples. r 7→ r 1+α , α ∈ (0, 1], and r 7→ (1 + r ) log (1 + r ) − r .
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Weak compactness in Lp , p ∈ [1, ∞]
The class CVP
Consider Φ ∈ CVP . Then
Φ(r )
is non-decreasing and concave,
r
Φ(r ) ≤ r Φ0 (r ) ≤ 2 Φ(r ),
r 7→
Φ(λr ) ≤ max {1, λ2 } Φ(r ),
0 ≤ (r + s) (Φ(r + s) − Φ(r ) − Φ(s)) ≤ 2 (r Φ(s) + s Φ(r )) ,
for r ≥ 0, s ≥ 0, and λ ≥ 0.
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Back to the discrete coagulation equation
Outline
1
Existence: Bounded coagulation kernels
2
Existence: Unbounded coagulation kernels
3
Weak compactness in Lp , p ∈ [1, ∞]
4
Back to the discrete coagulation equation
5
Existence: Unbounded coagulation kernels
(IMT)
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Back to the discrete coagulation equation
Existence revisited
Recall the following result (which excludes K (i, j) = i + j).
Theorem
Assume that K (i, j) ≤ Aα (ij)α , i, j ≥ 1, for some α ∈ [0, 1) and consider
f in = (fiin )i≥1 such that fiin ≥ 0, i ≥ 1. Then there is
f = (fi )i≥1 ∈ C([0, ∞); `10 ) ∩ L∞ (0, ∞, `11 )
such that, for i ≥ 1 and t > 0, (weak formulation)


Z t
i−1
∞
X
X
1
K (j, i − j)fj (s)fi−j (s) − fi (s)
K (i, j)fj (s) ds .
fi (t) = fiin +
2
0
j=1
j=1
Furthermore, kf (t)k1,1 ≤ kf in k1,1 for t ≥ 0.
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Back to the discrete coagulation equation
Existence revisited
Theorem
Assume that K (i, j) ≤ B(i + j), i, j ≥ 1, and consider f in = (fiin )i≥1 such
that fiin ≥ 0, i ≥ 1. Then there is
f = (fi )i≥1 ∈ C([0, ∞); `10 ) ∩ L∞ (0, ∞, `11 )
such that, for i ≥ 1 and t > 0,


Z t
i−1
∞
X
X
1
K (j, i − j)fj (s)fi−j (s) − fi (s)
K (i, j)fj (s) ds ,
fi (t) = fiin +
2
0
j=1
j=1
AND kf (t)k1,1 = kf in k1,1 for t ≥ 0.
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Back to the discrete coagulation equation
Existence revisited
Approximation (N ≥ 1): K N (i, j) = min {K (i, j)N} −→ existence of
f N = (fiN )i≥1 and
kf N (t)k1,1 = % := kf in k1,1 ,
t ≥0.
Componentwise compactness of (f N )N≥1 : there are Nk → ∞ and
f ∈ C([0, ∞); `10 ) such that
i ≥1, T > .
lim sup fiNk (t) − fi (t) = 0 ,
k →∞ t∈[0,T ]
Limiting behaviour of
∞
X
K Nk (i, j)fjNk ?
j=1
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Back to the discrete coagulation equation
Generalized moments
Recall that, if (ϑi )i≥1 is a sequence of real numbers, it (formally)
follows from the Fubini-Tonelli theorem that
∞
∞
1 X
d X
ϑ i fi =
ϑi+j − ϑi − ϑj K (i, j) fi fj .
dt
2
i=1
i,j=1
Lemma
If f in ∈ `11 is such that fiin ≥ 0 for i ≥ 1 and
∞
X
Φ(i)fiin < ∞
i=1
for some Φ ∈ CVP , then
∞
X
i=1
(IMT)
Φ(i)fiN (t)
≤e
2B%t
∞
X
Φ(i)fiin ,
t ≥0.
i=1
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Back to the discrete coagulation equation
Generalized moments: proof
Take ϑi = Φ(i). Then
∞
d X
Φ(i)fiN
dt
∞
=
i=1
≤
1X
[Φ(i + j) − Φ(i) − Φ(j)]K N (i, j)fiN fjN
2
i,j
∞
X
i,j
≤ 2B
[iΦ(j) + jΦ(i)]
B(i + j)fiN fjN
i +j
∞
X
jΦ(i)fiN fjN
i,j=1
∞
X
≤ 2B%
Φ(i)fiN .
i=1
Rigorous proof: truncation argument.
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Back to the discrete coagulation equation
Limit of
P
K N (i, j) fjN
i 7→ i ∈ L1 (N \ {0}; f in ) −→ there is Φ ∈ CVP,∞ such that
i 7→ Φ(i) ∈ L1 (N \ {0}; f in ) by the de la Vallée-Poussin theorem,
i.e.,
∞
X
MΦ :=
Φ(i)fiin < ∞ .
i=1
By the previous lemma,
∞
X
Φ(i)fiN (t) ≤ e2B%t MΦ ,
t ≥0.
i=1
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Back to the discrete coagulation equation
Limit of
P
K N (i, j) fjN
Fix i ≥ 1. Then, for i ≤ J ≤ N/2Bi,
X
∞ N
N
K (i, j) fj − K (i, j) fj j=1
J X
N
≤
−→ 0
K (i, j) fj − K (i, j) fj N→∞
j=1
+
∞
X

K N (i, j) fjN ≤ 2B
j=J+1
+
∞
X
j=J+1
(IMT)
∞
X

jfjN ≤ . . .
j=J+1

K (i, j) fj
≤ Ai
∞
X
j=J+1

jfj −→ 0
J→∞
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Back to the discrete coagulation equation
Limit of
P
K N (i, j) fjN
For i ≤ J ≤ N/2Bi,
∞
X
N
K (i, j)
fjN
≤ 2B
j=J+1
≤ 2B
≤
≤
(IMT)
∞
X
j=J+1
∞
X
j=J+1
∞
X
J
Φ(J)
jfjN
j
Φ(j)fjN
Φ(j)
Φ(j)fjN
j=1
J
e2B%t MΦ −→ 0 .
Φ(J)
J→∞
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Existence: Unbounded coagulation kernels
Outline
1
Existence: Bounded coagulation kernels
2
Existence: Unbounded coagulation kernels
3
Weak compactness in Lp , p ∈ [1, ∞]
4
Back to the discrete coagulation equation
5
Existence: Unbounded coagulation kernels
(IMT)
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Existence: Unbounded coagulation kernels
Approximation
Let K be a coagulation kernel and consider f in ∈ L11 (0, ∞), f in ≥ 0 a.e.
in (0, ∞).
“Truncation” parameter N ≥ 1. Define
K N (x, y ) = min {K (x, y ), N} ≤ N ,
x >0, y >0.
We have already shown that there exists a unique solution
f N ∈ C 1 ([0, ∞); L10 (0, ∞)) ∩ L∞ (0, ∞; L11 (0, ∞))
to the CSCE with coagulation kernel K N satisfying
f N (t, x) ≥ 0 a.e. in (0, ∞) ,
M0 (f N (t)) ≤ M0 (f in ) ,
M1 (f N (t)) = % := M1 (f in ) ,
for t ≥ 0.
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Existence: Unbounded coagulation kernels
The limit N → ∞
N
∂t f (t, x) =
1
2
x
Z
0
Z
−
K N (y , x − y ) f N (t, y ) f N (t, x − y ) dy
∞
K N (x, y ) f N (t, x) f N (t, y ) dy .
0
We have to identify the following limits:
f N as N → ∞ ,
Z ∞
K N (x, y ) f N (t, y ) dy as N → ∞ ,
Z0 x
K N (x − y , y ) f N (t, y ) f N (t, x − y ) dy as N → ∞ .
0
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Existence: Unbounded coagulation kernels
The limit N → ∞
Consider ϑ ∈ L∞ (0, ∞). Then, by Fubini’s theorem,
Z ∞Z x
K N (x − y , y ) f N (t, y ) f N (t, x − y ) dy ϑ(x) dx
Z0 ∞ Z0 ∞
=
K N (x, y ) ϑ(x + y ) f N (t, x) f N (t, y ) dydx
0
0
Z ∞Z ∞ N
K (x, y ) ϑ(x + y )
=
(1 + x)α f N (t, x)(1 + y )α f N (t, y ) dydx .
α (1 + y )α
(1
+
x)
0
0
Remark. If gn * g in L1 (0, ∞) and hn * h in L1 (0, ∞), then
(x, y ) 7−→ gn (x)hn (y ) converges weakly towards (x, y ) 7−→ g(x)h(y )in
L1 ((0, ∞) × (0, ∞)).
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Existence: Unbounded coagulation kernels
Compactness
Estimates:
f N (t, x) ≥ 0 a.e. in (0, ∞)
and
kf N (t)k1,1 ≤ kf in k1,1 ,
t ≥0.
which only implies that (f N )N≥1 is bounded in L∞ (0, ∞; L11 (0, ∞)) −→
no available compactness in the x-variable at this stage (6= discrete
setting).
Nevertheless, if
K (x, y ) ≤ A(1 + x)(1 + y ) ,
hen
N ∂t f (t)
1,0
(IMT)
≤
3A in 2
kf k1,1 ,
2
x >0, y >0
t ≥0.
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Existence: Unbounded coagulation kernels
Weak compactness in L1 (0, ∞)
Recall that f in ∈ L11 (0, ∞) which implies that
f in ∈ L1 (0, ∞) and x 7→ x ∈ L1 (0, ∞; f in dx) .
According to the de la Vallée-Poussin theorem there are two functions
Ψ, Φ ∈ CVP,∞ such that
Ψ(f in ) ∈ L1 (0, ∞) and x 7→ Φ(x) ∈ L1 (0, ∞; f in (x)dx) ,
that is,
Z
∞
in
Z
Ψ(f (x))dx < ∞ and
0
(IMT)
∞
Φ(x)f in (x)dx < ∞ .
0
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Existence: Unbounded coagulation kernels
Weak compactness in L1 (0, ∞)
Lemma
Given R > 0 and T > 0, there exists C(R, T ) > 0 such that
R
Z
Ψ(f N (t, x)) dx ≤ C(R, T ) ,
t ∈ [0, T ] ,
N≥1.
0
Corollary
Let T > 0. For each t ∈ [0, T ], (f N (t))N≥1 satisfies the assumptions of
the Dunford-Pettis theorem uniformly with respect to t ∈ [0, T ].
The control for large values of x is performed as for the DSCE.
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Existence: Unbounded coagulation kernels
Weak compactness in L1 (0, ∞): proof
Fix R > 0.
d
dt
Z
1
≤
2
Z
≤
Z
R
0
R
Ψ(f N ) dx
0
R
R
K N (x − y , y )f N (x − y ) Ψ0 (f N (x)) dx f N (y ) dy
y
R
Z
K N (x − y , y )f N (x − y )f N (y ) dy Ψ0 (f N (x)) dx
0
Z
0
x
Z
Z
≤
0
≤ 2
R
h
i
K N (x − y , y ) Ψ(f N (x − y )) + Ψ(f N (x)) dx f N (y ) dy
y
sup
(0,R)×(0,R)
(IMT)
{K } kf N k1,0
Z
R
Ψ(f N ) dx
0
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Existence: Unbounded coagulation kernels
The end
Previous estimates + variant of the Arzelà-Ascoli theorem −→
there are a subsequence (f Nk )k ≥1 and a non-negative function
f ∈ C([0, ∞; w − L1 (0, 1)) such that
Z ∞ N
f k (t, x) − f (t, x) ϑ(x) dx = 0
lim sup k →∞ t∈[0,T ]
0
for all T > 0 and ϑ ∈ L∞ (0, ∞).
If K (x, y ) ≤ Aα (1 + x)α (1 + y )α for some α ∈ [0, 1), then f is a
weak solution to the CSCE and M1 (f (t)) ≤ M1 (f in ) for all t ≥ 0.
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Existence: Unbounded coagulation kernels
Weak solution
Definition
A non-negative function
f ∈ C([0, ∞); w − L1 (0, 1)) ∩ L∞ (0, ∞; L11 (0, ∞))
is a weak solution to the CSCE if
Z ∞
f (t, x) − f in (x) ϑ(x) dx
0
Z t Z ∞Z ∞
1
[ϑ(x + y ) − ϑ(x) − ϑ(y )]
=
2 0 0
0
K (x, y )f (s, x)f (s, y ) dydxds
for all t ≥ 0 and ϑ ∈ L∞ (0, ∞).
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Existence: Unbounded coagulation kernels
Comments
Non-increase of total mass cannot be improved to conservation of
mass in general.
Strong continuity in L1 (0, ∞) can be proved afterwards.
Uniqueness and continuous dependence have to be proved
separately (6= semigroup approach).
Uniqueness under additional assumptions and weak stability.
More complicated arguments are required to handle the cases
K (x, y ) = x + y and K (x, y ) = xy .
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