Weak compactness techniques for
coagulation equations.
III. Gelation, uniqueness, and self-similarity
Philippe Laurençot
Institut de Mathématiques de Toulouse
29.07.2013
(IMT)
29.07.2013
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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
0
Z
−
∞
K (x, y ) f (t, x) f (t, y ) dy .
0
Initial condition: f in
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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Notation
For µ ∈ R,
L1µ (0, ∞)
:=
Z
1
f ∈ L (0, ∞) : kf k1,µ =
∞
(1 + x )|f (x)| < ∞ .
µ
0
Z
Mµ (f ) :=
∞
x µ f (x) dx .
0
Initial condition: f in ∈ L11 (0, ∞), f in ≥ 0 a.e. in (0, ∞).
(IMT)
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Outline
1
Gelation/Runaway growth
2
Uniqueness
3
Self-similarity
4
Dynamical approach to steady states
(IMT)
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Gelation/Runaway growth
Outline
1
Gelation/Runaway growth
2
Uniqueness
3
Self-similarity
4
Dynamical approach to steady states
(IMT)
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Gelation/Runaway growth
Mass conservation/gelation
If K (x, y ) ≤ A(2 + x + y ), there is at least a mass-conserving
solution.
If K (x, y ) ≥ (xy )λ/2 for some λ ∈ (1, 2], then gelation occurs, that
is, there is Tgel ∈ [0, ∞) such that
M1 (f (t)) < M1 (f in ) for T > Tgel .
Remark. Though known/conjectured since 1980 and supported by a
few explicit solutions, the occurrence of gelation was only established
for all initial data in 2003.
(IMT)
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Gelation/Runaway growth
Gelation
Assumption. K (x, y ) ≥ (xy )λ/2 for some λ ∈ (1, 2].
Proposition
Let ξ : [0, ∞) −→ [0, ∞) be a non-decreasing differentiable function
satisfying ξ(0) = 0 and
Z ∞
Iξ :=
ξ 0 (A) A−1/2 dA < ∞ .
0
For t > 0,
Z t Z
x
0
(IMT)
0
2
∞
λ/2
ξ(x) f (s, x) dx
ds ≤ Iξ2 M1 (f in ) .
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Gelation/Runaway growth
Gelation
Aim: Take ξ(x) = x (2−λ)/2 so that, for t > 0,
Z
t
M1 (f (s)) ds =
0
x
0
2
∞
Z t Z
2
λ/2
ξ(x) f (s, x) dx
0
ds ≤ Iξ2 M1 (f in ) .
Then M1 (f (t)) −→ 0 as t → ∞.
However
Z
∞
0
−1/2
ξ (A) A
Z
dA < ∞ but
1
ξ 0 (A) A−1/2 dA = ∞ .
0
(2−λ)/2
Take instead ξ(x) = (x − 1)+
(IMT)
1
.
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Gelation/Runaway growth
Proof of the proposition
Fix A > 0 and take ϑ(x) = min {x, A} in
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
As ϑ(x + y ) − ϑ(x) − ϑ(y ) ≤ 0,
Z t Z
x
0
2
∞
A
λ/2
f (s, x) dx
ds ≤
M1 (f in )
.
A
Here, A is a free parameter. Integrate with respect to A.
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Uniqueness
Outline
1
Gelation/Runaway growth
2
Uniqueness
3
Self-similarity
4
Dynamical approach to steady states
(IMT)
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Uniqueness
Direct approach
Theorem
Assume that there is a non-negative subadditive function ϕ
(ϕ(x + y ) ≤ ϕ(x) + ϕ(y ), x > 0, y > 0), such that
K (x, y ) ≤ ϕ(x)ϕ(y ) ,
x >0, y >0.
Then there is at most one solution
f ∈ C([0, T ]; L1 (0, ∞; ϕ(x)dx) ∩ L1 (0, T ; L1 (0, ∞; ϕ(x)2 dx) ,
T >0,
to the CSCE.
Remark. ϕ(x) ≤ x 1/2 −→ uniqueness in L11 (0, ∞).
(IMT)
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Uniqueness
Proof
=
Z ∞
d
|f1 − f2 |ϕ dx
dt 0
Z Z
1 ∞ ∞
K (x, y )(f1 + f2 )(y )(f1 − f2 )(x)Θ(x, y ) dydx
2 0
0
with
Θ(x, y ) := [(ϕσ)(x + y ) − (ϕσ)(x) − (ϕσ)(y )] ,
σ := sign(f1 − f2 ) .
(f1 − f2 )(x)Θ(x, y ) = |(f1 − f2 )(x)|σ(x)Θ(x, y )
h
i
=|(f1 − f2 )(x)| (ϕσ)(x + y )σ(x) − ϕ(x)σ(x)2 − (ϕσ)(y )σ(x)
≤|(f1 − f2 )(x)| [ϕ(x + y ) − ϕ(x) + ϕ(y )]
≤2ϕ(y )|(f1 − f2 )(x)| .
(IMT)
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Uniqueness
Proof
Z ∞
d
|f1 − f2 |ϕ dx
dt 0
Z ∞Z ∞
≤
K (x, y )(f1 + f2 )(y )|(f1 − f2 )(x)|ϕ(y ) dydx
0
0
Z ∞Z ∞
≤
ϕ(x)ϕ(y )2 (f1 + f2 )(y )|(f1 − f2 )(x)| dydx
0
0
Z ∞
Z ∞
2
=
ϕ(y ) (f1 + f2 )(y ) dy
ϕ(x)|(f1 − f2 )(x)| dx ,
0
0
and the conclusion follows by integration.
(IMT)
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Uniqueness
Alternative approach: “Wasserstein distance”
Constant kernel: K = 2. Define
Z ∞
F (t, x) :=
f (t, y ) dy ,
t >0, x >0.
x
Then it solves
Z x
∂t F (t, x) =
f (t, y )F (t, x − y ) dy − F (t, 0)F (t, x) ,
t >0, x >0,
0
and
t 7−→ kF1 (t) − F2 (t)k1,0 is non-increasing,
F1in ≤ F2in implies F1 (t) ≤ F2 (t), t ≥ 0.
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Uniqueness
Alternative approach
General setting: homogeneous kernel K (x, y ) = x α y β + x β y α ,
λ := α + β ∈ (0, 2].
Theorem
Let T > 0. There is at most one solution f ∈ C([0, T ); w − L1λ (0, ∞)) to
the CSCE.
Remark. If λ ∈ (0, 1], global existence. If λ ∈ (1, 2], local existence, up
to the gelation time.
(IMT)
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Uniqueness
Alternative approach
General setting: homogeneous kernel K (x, y ) = x α y β + x β y α ,
λ := α + β ∈ (0, 2].
Z ∞
Fi (t, x) :=
fi (t, y ) dy , t > 0 , x > 0 , i = 1, 2 .
x
Z
R(t, x) :=
x
z λ−1 sign(F1 − F2 )(t, z) dz ,
t >0, x >0.
0
d
dt
(IMT)
Z
0
∞
x λ−1 |(F1 − F2 )(t, x)| dx ≤
1
(A1 (t) + A2 (t)) ,
2
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Uniqueness
Alternative approach
∞Z ∞
Z
A1 (t) :=
0
h
i
K (x, y ) (x + y )λ−1 − x λ−1
0
(f1 + f2 )(t, y )|(F1 − F2 )(t, x)| dydx
λ−1
λ−1 K (x, y ) (x + y )
−x
≤ Cx λ−1 y λ .
Z
A1 (t) ≤ C
0
(IMT)
∞
y λ (f1 + f2 )(t, y ) dy
Z
∞
x λ−1 |(F1 − F2 )(t, x)| dx .
0
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Uniqueness
Alternative approach
∞Z ∞
Z
∂x K (x, y ) [R(t, x + y ) − R(t, x) − R(t, y )]
A2 (t) :=
0
0
(f1 + f2 )(t, y )(F1 − F2 )(t, x) dydx
|∂x K (x, y )| |R(t, x + y ) − R(t, x) − R(t, y )| ≤ Cx λ−1 y λ .
Z
A2 (t) ≤ C
0
(IMT)
∞
y λ (f1 + f2 )(t, y ) dy
Z
∞
x λ−1 |(F1 − F2 )(t, x)| dx .
0
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Uniqueness
Alternative approach
Z
d
dt
Z
∞
0
∞
≤ C
0
x λ−1 |(F1 − F2 )(t, x)| dx
Z ∞
y λ (f1 + f2 )(t, y ) dy
x λ−1 |(F1 − F2 )(t, x)| dydx ,
0
and the conclusion follows by integration.
(IMT)
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Self-similarity
Outline
1
Gelation/Runaway growth
2
Uniqueness
3
Self-similarity
4
Dynamical approach to steady states
(IMT)
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Self-similarity
Dynamic scaling assumption
Outcome of the Smoluchowski coagulation equation:
Decay of the total number of particles M0 (f (t)) to zero as t → ∞.
Either conservation of the total mass M1 (f (t)) for all times or
occurrence of gelation at a finite time Tgel .
Dynamical scaling assumption:
1
x
f (t, x) ∼
ϕ
s(t)γ
s(t)
as t → ∞ or t → Tgel ,
where
s(t) denotes the mean size of the particles at time t > 0 and is
thus an increasing function,
the function ϕ denotes the scaling profile,
and γ ∈ R.
(IMT)
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Self-similarity
Dynamical scaling assumption
Dynamical scaling assumption:
1
x
f (t, x) ∼
ϕ
s(t)γ
s(t)
as t → ∞ or t → Tgel .
Alternatively, stationary behaviour after rescaling:
s(t)γ f (t, xs(t)) ∼ ϕ(x) as t → ∞ or t → Tgel .
Preliminary question. Are there such solutions to the CSCE? This
obviously requires some homogeneity of the coagulation kernel K , that
is,
K (ax, ay ) = aλ K (x, y ) ,
x >0, y >0 a>0.
for some λ ∈ R.
(IMT)
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Self-similarity
Coagulation kernel K : examples
K (x, y ) =
K (x, y ) = 2
x
1/3
1
1
+y
+
,
x 1/3 y 1/3
λ=0,
1/3
λ=0
K (x, y ) = (ax + b)(ay + b) , a > 0 , b ≥ 0 ,
λ = 2 if b = 0 ,
3
K (x, y ) =
x 1/3 + y 1/3
,
λ=1,
2 1/3
K (x, y ) =
x 1/3 + y 1/3
− y 1/3 ,
λ=1,
x
K (x, y ) = x α y β + x β y α ,
(IMT)
α ≤ 1,β ≤ 1 ,
λ=α+β.
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Self-similarity
Self-similar solutions
Homogeneous coagulation kernel:
K (ax, ay ) = aλ K (x, y ) ,
x >0, y >0 a>0.
for some λ ∈ R.
We look for a solution of the form
1
x
ϕ
,
f (t, x) =
s(t)γ
s(t)
t ∈ [0, ∞) or t ∈ [0, Tgel ) .
Inserting this ansatz in the CSCE gives an ordinary differential
equation for s and a nonlinear integro-differential equation for ϕ:
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Self-similarity
Self-similar solutions
Ordinary differential equation for s:
sγ−λ−2
ds
=w ∈R.
dt
Nonlinear integro-differential equation for ϕ:
Z
1 x
w (γϕ(x) + x∂x ϕ(x)) = −
K (x − y , y )ϕ(x − y )ϕ(y ) dy
2 0
Z ∞
+
K (x, y )ϕ(x)ϕ(y ) dy .
0
Mass-conserving self-similar solutions: γ = 2.
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Self-similarity
Mean size s(t)
Ordinary differential equation for s:
sγ−λ−2
ds
=w ∈R.
dt
The assumed monotonicity of s requires w > 0. Thus
if γ > λ + 1, then
1/(γ−λ−1)
s(t) = s(0)γ−λ−1 + w(γ − λ − 1)t
,
if γ = λ + 1, then s(t) = s(0) ewt ,
if γ < λ + 1, then
s(t) = s(0)
(IMT)
γ−λ−1
− w(1 + λ − γ)t
−1/(1+λ−γ)
.
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Self-similarity
Mean size s(t)
Ordinary differential equation for s:
sγ−λ−2
ds
=w ∈R.
dt
s(t) → ∞ as t → ∞ if γ ≥ λ + 1.
s(t) → ∞ as t → T∗ if γ < λ + 1 for some finite T∗ > 0.
(IMT)
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Self-similarity
The profile equation
1
w (γϕ(x) + x∂x ϕ(x)) = −
2
Z
+
Z
x
K (x − y , y )ϕ(x − y )ϕ(y ) dy
0
∞
K (x, y )ϕ(x)ϕ(y ) dy .
0
It cannot be seen as an initial value problem as the value of ϕ(x)
depends on that of ϕ(y ) for all y ∈ (0, ∞) and not only for y ∈ (0, x).
(IMT)
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Self-similarity
Explicit solutions
K (x, y ) = 2: λ = 0, γ = 1 + ρ ∈ (λ + 1, 2] and
ϕρ (x) =
∞
X
(−1)k −1
k =1
Γ(ρk )
x ρk −1 ,
x >0.
In particular, ϕ1 (x) = e−x , x > 0.
K (x, y ) = x + y : λ = 1, γ = λ + 1, w = (1 + ρ)/ρ ∈ [2, ∞), and
ϕρ (x) =
∞
X
(−1)k −1
1
πx (w+1)/w
k =1
k!
k
Γ 1+k −
w
sin
k π k /w
x
w
In particular,
1 e−x/2
,
ϕ1 (x) = √
2π x 3/2
(IMT)
x >0.
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Self-similarity
Explicit solutions
K (x, y ) = xy : λ = 2, γ = (2ρ + 3)/(ρ + 1) ∈ [5/2, λ + 1), , and
ϕρ (x) =
∞
1 X (−1)k −1
Γ (1 + (γ − 2)k ) sin ((3 − γ)k π)x (3−γ)k
k!
πx 2
k =1
In particular,
1 e−x/2
ϕ1 (x) = √
,
2π x 5/2
(IMT)
x >0.
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Self-similarity
The profile equation
K (x, y ) = (xy )λ/2 , 0 < λ < 1, γ = 2, w = 1
Look for a non-negative solution ϕ with M1 (ϕ) < ∞ to
Z
1 x
K (x − y , y )ϕ(x − y )ϕ(y ) dy
2ϕ(x) + xϕ0 (x) = −
2 0
Z ∞
+
K (x, y )ϕ(x)ϕ(y ) dy .
0
Assume that ϕ is smooth at x = 0. Taking x = 0 and using K (0, y ) = 0
entails ϕ(0) = 0. Dividing by x and letting x → 0 next gives ϕ0 (0) = 0.
Also,
Z ∞
2
K (x, y )
0
ϕ (x) + q(x)ϕ(x) ≤ 0 ,
q(x) := −
ϕ(y ) dy .
x
x
0
(IMT)
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Self-similarity
Initial singularity of the profile
K (x, y ) = (xy )λ/2 , 0 < λ < 1, γ = 2, w = 1
Ansatz: ϕ(x) ∼ Ax −τ as x → 0.
Formal computation: τ = 1 + λ with an explicit constant A = A(λ).
Problem: y 7→ y λ/2 ϕ(y ) is not integrable on (0, 1) and thus
Z ∞
Z ∞
λ/2
K (x, y )ϕ(x)ϕ(y ) dy = x ϕ(x)
y λ/2 ϕ(y ) dy = ∞ .
0
0
Alternative formulation.
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Self-similarity
Alternative formulation of the CSCE
In the weak formulation, take ϑ(x) = xη(x) −→
Z x Z ∞
y K (y , z) f (t, y ) f (t, z) dzdy .
x∂t f (t, x) = −∂x
0
x−y
Total mass conservation is equivalent to no flux of mass as
x → ∞.
Useful formulation to construct finite volume numerical schemes.
(IMT)
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Self-similarity
Alternative equation for the profile
2
Z
x
Z
∞
y K (y , z) ϕ(y ) ϕ(z) dzdy ,
x ϕ(x) =
0
x >0.
x−y
Well-defined for the expected singularity.
Still not an initial value problem.
ϕ = 0 is always a solution!
Existence: Contraction mapping principle? Contraction?
Existence: Schauder fixed point? Strong compactness?
Existence: Tychonov fixed point? Weak compactness?
(IMT)
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Dynamical approach to steady states
Outline
1
Gelation/Runaway growth
2
Uniqueness
3
Self-similarity
4
Dynamical approach to steady states
(IMT)
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Dynamical approach to steady states
Fixed point theorems
Theorem (Brouwer)
Let Y be a non-empty, compact, and convex subset of Rm , m ≥ 1, and
g ∈ C(Y ; Rm ) such that g(Y ) ⊂ Y . Then g has a fixed point in Y , that
is, there is x0 ∈ Y such that g(x0 ) = x0 .
Remark. No uniqueness statement.
Theorem (Schauder)
Let E be a Banach space and Y a non-empty closed convex subset of
E. If g ∈ C(Y ; E) is such that g(Y ) ⊂ Y and g(Y ) is relatively
compact, then g has a fixed point in Y , that is, there is x0 ∈ Y such
that g(x0 ) = x0 .
(IMT)
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Dynamical approach to steady states
Fixed point theorems
Theorem (Tychonov)
Let E be a locally convex real vector space and Y a non-empty closed
compact and convex subset of E. If g ∈ C(Y ; E) is such that
g(Y ) ⊂ Y , then g has a fixed point in Y , that is, there is x0 ∈ Y such
that g(x0 ) = x0 .
Remark. A Banach space endowed with its weak topology σ(E, E 0 ) is
locally convex.
(IMT)
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Dynamical approach to steady states
Dynamical systems
Definition
Let X be a topological space and J an interval of R containing 0. A
continuous map Ψ : J × X → X is a (continuous) dynamical system if
Ψ(0, x) = x for all x ∈ X ,
Ψ(s, Ψ(t, x)) = Ψ(s + t, x) for all x ∈ X , s ∈ J, and t ∈ J such that
s + t ∈ J.
Definition
Let X be a topological space, J an interval of R containing 0 and
Ψ : J × X → X is a (continuous) dynamical system. A subset M of X is
said to be positively invariant by Ψ if Ψ(t, x) ∈ M for all t ∈ J ∩ [0, ∞)
and x ∈ M.
(IMT)
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Dynamical approach to steady states
Dynamical approach to steady states
Theorem
Let m ≥ 1 and Ψ : [0, ∞) × Rm → Rm a (continuous) dynamical system
with a positively invariant region M which is a non-empty compact and
convex subset of Rm . Then there is x0 ∈ M such that Ψ(t, x0 ) = x0 for
all t ≥ 0.
(IMT)
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Dynamical approach to steady states
Proof: Periodic solutions
Fix T > 0 and define g : Rm → Rm by
g(x) = Ψ(T , x) ,
x ∈ Rm .
Then g ∈ C(M; Rm ) and g(M) ⊂ M. By Brouwer’s theorem, there
exists xT ∈ M such that g(xT ) = xT . This implies that t 7→ Ψ(t, xT ) is
T -periodic. Consequently,
PT := {x ∈ M : t 7→ Ψ(t, x) is T − periodic. } =
6 ∅.
(IMT)
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Dynamical approach to steady states
Proof: Stationary solutions
For all T > 0,
PT := {x ∈ M : t 7→ Ψ(t, x) is T − periodic. } =
6 ∅.
Furthermore, PT is clearly a compact set as a closed subset of M and
PT /2 ⊂ PT . Therefore,
P0 :=
\
P2−k 6= ∅
k ≥1
and any element x0 of P0 satisfies Ψ(t, x0 ) = x0 for all t ≥ 0.
(IMT)
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Dynamical approach to steady states
Existence of self-similar solutions
Consider the evolution equation (g(t, x) ∼ xf (t, x) up to a rescaling):
Z xZ ∞
K (y , z)
∂t g(t, x) = ∂x xg(t, x) −
g(t, y ) g(t, z) dzdy .
z
0
x−y
Discrete equation: Let n ≥ 1 and N = n2 . Set
i −1
i j
n
n
vi :=
, Ki,j := K
,
, 1 ≤ i, j ≤ N ,
n
n n
n
=0,
vN+1
and, for 1 ≤ i ≤ N and g n := (gjn )1≤j≤N ,
i−1
n
N−i
n
X Ki−j,j
X Ki,j
dgin
n
n
n
= Fin (q n ) := vi+1
gi+1
− vin gin +
gi−j
gjn −
gngn .
dt
j
j i j
j=1
(IMT)
j=1
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Dynamical approach to steady states
Discretization
For 1 ≤ i ≤ N and g n := (gjn )1≤j≤N ,
i−1
n
N−i
n
X Ki,j
X Ki−j,j
dgin
n
n
n
= Fin (q n ) := vi+1
gi−j
gngn .
gjn −
gi+1
− vin gin +
dt
j
j i j
j=1
j=1
Define


N


X
Y n := y = (yi )1≤i≤N ∈ RN : yi ≥ 0 , 1 ≤ i ≤ N ,
yi = % .


j=1
Continuous dynamical system on Y n and Y n is positively invariant as
well as convex and compact
−→ stationary solution q n = (qin )1≤i≤N ∈ Y n .
(IMT)
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Dynamical approach to steady states
Existence
For n ≥ 1, define a bounded measure Q n (dx) by
n
Q (dx) :=
N
X
qin δx=i/n ,
hQ n (dx), 1i = % .
i=1
Moment estimates:
suphQ n (dx), x σ i < ∞ for σ > λ − 1 ,
n≥1
which is compatible with the expected singularity as x → 0.
“Integrability estimates”: for p ∈ [1, 1/λ),
1
n
n
Z (x) := n Q (dy ), y −
1[x,x+1/n] (y ) ∈ Lp (0, ∞) .
n
(IMT)
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Dynamical approach to steady states
Existence
“Integrability estimates”:
Z n (x)
:=
“ =00
1
1[x,x+1/n] (y )
n Q n (dy ), y −
n
Z x+1/n 1
n
y−
Q n (dy )
n
x
Convergence towards a measurable function Q. The sought for
profile is ϕ(x) = Q(x)/x.
Exponential moments.
(IMT)
29.07.2013
45 / 46
Dynamical approach to steady states
Comments
There is still no proof of ϕ(x) ∼ Ax −1−λ as x → 0 (only lower and
upper bounds).
γ = 2: Existence of self-similar solutions for a large class of
coagulation kernels, even unbounded as x → 0 or y → 0.
γ ∈ (1 + λ, 2): Existence of self-similar solutions for some
coagulation kernels.
Uniqueness? Stability?
(IMT)
29.07.2013
46 / 46