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Section 3: Infinite-Dimensional State Space
Contents
I Abstract Cauchy Problems.
I Discrete-Size Models of Coagulation-Fragmentation Processes.
I Linear ACPs and Strongly Continuous Semigroups
I Semilinear ACPs.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
ACPs
I State space X is now an infinite-dimensional Banach space with norm k · k.
I Aim is to express evolution equations, in operator form, as ODEs posed in
X.
I We shall consider only problems of the type
u0(t) = L(u(t)) + N (u(t)), t > 0,
◦
u(0) = u,
(†)
where L : X ⊇ D(L) → X and N : X → X are, respectively, linear and
nonlinear operators, with D(L) a linear subspace of X.
I In (†), the derivative is defined in terms of the norm on X, i.e.
u(t + h) − u(t)
0
− u (t))
→ 0 as h → 0,
h
and a solution u : [0, ∞) → X is sought.
I The IVP (†) is usually called a (semilinear) abstract Cauchy problem (ACP).
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Discrete C–F Equations
I System of particles that can coagulate, to form larger particles, or fragment
into smaller particles.
I C–F processes arise in many areas of science and engineering;
e.g.astrophysics, blood clotting, colloidal chemistry and polymer science,
molecular beam epitaxy, mathematical ecology.
I Assume system consists of a large number of clusters (or mers) that can
coagulate to form larger clusters or fragment into smaller clusters.
I If each cluster of size n (n-mer) is composed of n identical fundamental
units (monomers), the mass of each cluster is simply an integer multiple of the
mass of a monomer.
I By appropriate scaling, each monomer can be assumed to have unit mass.
This then leads to a so-called discrete model of coagulation-fragmentation,
with discrete indicating that cluster mass is a discrete variable which, in view
of the above, can be assumed to take positive integer values.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Discrete C–F Equations
I We consider the following kinetic equation describing the time-evolution of
the clusters:
d
dt
un(t) = −anun(t) +
∞
X
aj bn,j uj (t)
j =n+1
+
◦
n−1
1X
2
un(0) = un ,
kn−j,j un−j (t)uj (t) −
j =1
∞
X
j =1
(n = 1, 2, 3, ...) ,
I un(t) is the concentration of n-mers at time t
I an is the net rate of break-up of an n-mer.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
kn,j un(t)uj (t) ,
Discrete C–F Equations
I bn,j gives the average number of n-mers produced upon the break-up of a
j-mer
I kn,j = kj,n represents the coagulation rate of an n-mer with a j-mer.
I We shall assume for simplicity that kn,j ≡ k.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Discrete C–F Equations
I The total mass in the system at time t is given by
M (t) =
∞
X
nun(t) ,
n=1
and for mass to be conserved we require
j−
X1
nbn,j = j ,
(j = 2, 3, . . .).
n=1
On using this condition together with the C–F equation, a formal calculation
shows that Ṁ (t) = 0.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Discrete Binary C–F Equations
I When the fragmentation process is binary, we have bn,j = bj−n,j and the
number of clusters produced in any fragmentation event is always two, so that
Pj−1
n=1 bn,j = 2 .
I The C–F equation the becomes
d
dt
un(t) = −un(t)
n−
X1
j =1
+
n−1
1X
2
j =1
Fj,n−j + 2
∞
X
Fn,j−nuj (t)
j =n+1
kn−j,j un−j (t)uj (t) −
∞
X
kn,j un(t)uj (t) ,
j =1
I Fn,j = an+j bn,n+j /2 is a symmetric function of n and j that gives the net
rate at which (n + j)-mers break into n-mers and j-mers.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
ACP Formulation of Discrete C–F Equations
I Define u(t) = (u1(t), u2(t), . . . , uj (t), . . .) so that, for each t ≥ 0, u(t)
is a sequence-valued function of t.
I We seek a function u, defined on [0, ∞), that takes values in an infinitedimensional state space consisting of sequences.
I The state space that is most often used due to its physical relevance is the
Banach space `11 discussed in Example 1.4.
P∞
1
I The `1-norm of a non-negative element f , given by j =1 jf (j), represents
the total mass of the system.
I Similarly, the `10-norm of such an f gives the total number of particles in the
system.
I Note that `11 is continuously imbedded in `10.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
ACP Formulation of Discrete C–F Equations
I The sequence-valued function u will be required to satisfy an ACP of the
form
◦
u0(t) = L(u(t)) + N (u(t)), u(0) =u,
where L and N are appropriately defined operator versions of the respective
mappings
fn → −anfn+
∞
X
j =n+1
aj bn,j fj and fn →
n−1
kX
2
fn−j fj −k
j =1
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
∞
X
j =1
fnfj , (n ∈ N)
Linear Evolution Equations in Infinite-Dimensions
I Suppose the ACP involves only the linear operator L; i.e. it takes the form
u0(t) = L(u(t)), t > 0,
◦
u(0) = u .
I A function u : [0, ∞) → X is said to be a strong (or strict or classical)
solution to this linear ACP if
(i) u is continuous on [0, ∞);
(ii) u is continuously differentiable on (0, ∞);
(iii) u(t) ∈ D(L) for each t > 0;
(iv) the equations in the ACP are satisfied.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
L ∈ B(X)
I When L ∈ B(X) the unique strong solution is
u(t) = e
tL ◦
u,
where the operator exponential is defined by
etL = I + tL +
t2L2
2!
+
t3L3
3!
+ ··· ,
with I denoting the identity operator on X.
I This infinite series of bounded, linear operators on X always converges in
B(X) to a bounded, linear operator on X.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
I Moreover,
e
0L
= I; e
sL tL
e
=e
◦
( s + t) L
I The function φ(t, u) = e
dynamical system on X.
for all s, t ∈ R; e
◦
tL u
tL ◦
◦
u→u in X as t → 0.
defines a continuous, infinite-dimensional
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Unbounded L : The Hille-Yosida Theorem
I In many applications, L ∈
/ B(X).
I Einar Hille and Kôsaku Yosida, simultaneously and independently, proved a
theorem (the Hille–Yosida theorem) that forms the cornerstone of the Theory
of Strongly Continuous Semigroups of Operators.
I Since then, there has been a great deal of research activity in the theory
and application of semigroups of operators.
I The Hille–Yosida theorem was extended in 1952 to a result that completely
characterises the operators L that generate so-called strongly continuous
semigroups on a Banach space X.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Unbounded L : The Hille-Yosida Theorem
Definition 3.1. Let {S(t)}t≥0 be a family of bounded linear operators on
a complex Banach space X. Then {S(t)}t≥0 is a strongly continuous
semigroup (or C0- semigroup in B(X)) if the following conditions are satisfied.
S1. S(0) = I, where I is the identity operator on X.
S2. S(t)S(s) = S(t + s) for all t, s ≥ 0.
S3. S(t)f → f as t → 0+ for all f ∈ X.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
I Associated with each strongly continuous semigroup {S(t)}t≥0 is a unique
linear operator L defined by
Lf := lim
h→0+
S(h)f − f
h
,
D(L) :=
f ∈ X : lim
h→0+
I The operator L is called the infinitesimal generator
{S(t)}t≥0.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
S(h)f − f
h
exists in X
of the semigroup
Unbounded L : The Hille-Yosida Theorem
Definition 3.2 Let L : X ⊇ D(L) → X be a linear operator.
(i) The resolvent set, ρ(L), of L is the set of complex numbers
ρ(L) := {λ ∈ C : R(λ, L) := (λI − L)−1 ∈ B(X)};
R(λ, L) is called the resolvent operator of L (at λ).
(ii) L is a closed operator (or L is closed) if whenever (fn)∞
n=1 ⊂ D(L) is
such that fn → f and Lfn → g in X as n → ∞, then g ∈ D(L) and
Lf = g.
(iii) An operator L1 : X ⊃ D(L1) → X is an extension of L, written L ⊂ L1,
if D(L) ⊂ D(L1) and Lf = L1f for all f ∈ D(L). The operator L is
closable if it has a closed extension, in which case the closure L of L is
defined to be smallest closed extension of L.
(iv) L is densely defined if D(L) = X.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Unbounded L : The Hille-Yosida Theorem
Theorem 3.1 (Some Semigroup Results) Let {S(t)}t≥0 ⊂ B(X) be a
strongly continuous semigroup with infinitesimal generator L. Then
(i) S(t)f → S(t0)f in X as t → t0 for any t0 > 0 and f ∈ X;
(ii) there are real constants M ≥ 1 and ω such that
kS(t)k ≤ M eωt for all t ≥ 0;
(‡)
(iii) f ∈ D(L) ⇒ S(t)f ∈ D(L) for all t > 0 and
d
dt
S(t)f = LS(t)f = S(t)Lf for all t ≥ 0 and f ∈ D(L);
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
(iv) the infinitesimal generator L is closed and densely defined.
I We shall write L ∈ G(M, ω; X) when L is the infinitesimal generator of a
strongly continuous semigroups of operators, satisfying (‡) on a Banach space
X.
I When L ∈ G(1, 0; X) then it generates a strongly continuous semigroup of
contractions on X.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Unbounded L : The Hille-Yosida Theorem
Theorem 3.2
(Hille-Yosida) The operator L is the infinitesimal generator of a strongly
continuous semigroup of contractions on X if and only if (i) L is a closed,
linear and densely-defined operator in X; (ii) λ ∈ ρ(L) for all λ > 0; (iii)
kR(λ, L)k ≤ 1/λ for all λ > 0.
(Hille-Yosida-Phillips-Miyadera-Feller) L ∈ G(M, ω; X) if and only if (i) L
is a closed, linear and densely-defined operator in X; (ii) λ ∈ ρ(L) for all
λ > ω; (iii) k(R(λ, L))nk ≤ M/(λ − ω)n for all λ > ω, n = 1, 2, . . ..
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Unbounded L : Existence-Uniqueness Result
We can now state the following existence/uniqueness theorem for the linear
ACP
◦
u0(t) = L(u(t)), t > 0; u(0) =u ∈ D(L).
Theorem 3.3 Let L be the infinitesimal generator of a strongly continuous
semigroup {S(t)}t≥0 ⊂ B(X). Then the linear ACP has one and only one
◦
u
strong solution u : [0, ∞) → X and this is given by u(t) = S(t) .
It can be shown that S(t) can be interpreted as the exponential etL if we define
the latter by
etLf := lim (I − tL/n)−n, f ∈ X.
n→∞
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
The Kato-Voigt Perturbation Theorem
I The Hille-Yosida Theorem can be difficult to apply.
I To get round this, perturbation results are often used.
I Basic idea is to treat, if possible, the linear operator governing the dynamics
of the system as the sum of two linear operators, say A + B, where A is
an operator which can easily be shown to generate a strongly continuous
semigroup {SA(t)}t≥0 on a Banach space X, and B is regarded as a
perturbation of A.
I The question then is to identify sufficient conditions on B which will
guarantee that A + B (or some extension of A + B) also generates a strongly
continuous semigroup on X.
I A number of perturbation results of this type have been established.
I We consider just one of these, namely the Kato-Voigt Perturbation theorem,
but only for the specific case when the state space is the Banach space `1µ of
Example 1.4.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
The Kato-Voigt Perturbation Theorem
I Non-negative elements in `1µ are taken to be sequences
f = (f1, f2, . . . , fn, . . .) with fj ≥ 0 for all j.
I We write f ≥ 0.
I An operator T : `1µ ⊇ D(T ) → `1µ, is said to be non-negative if T f ≥ 0 for
all non-negative f ∈ D(T ).
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
The Kato-Voigt Perturbation Theorem
Theorem 3.4 Let A : `1µ ⊇ D(A) → `1µ and B : `1µ ⊇ D(B) → `1µ have the
following properties.
(i) A is the infinitesimal generator of a semigroup of contractions {SA(t)}t≥0
on `1µ, with SA(t) ≥ 0 for all t ≥ 0.
(ii) B is non-negative and D(B) ⊇ D(A).
(iii) For each non-negative f in D(A),
∞
X
j µ(Af + Bf )j ≤ 0.
j =1
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Then there exists a strongly continuous semigroup of contractions, {S(t)}t≥0,
on `1µ satisfying the Duhamel equation
Z
t
S(t − s)BSA(s)f ds, f ∈ D(A).
S(t)f := SA(t)f +
0
Each S(t) is non-negative and the infinitesimal generator of the semigroup is
an extension L of A + B.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Semilinear Infinite-Dimensional Evolution Equations
I To conclude, we return to the semilinear ACP
u0(t) = L(u(t)) + N (u(t)), t > 0,
◦
u(0) = u .
(†)
I Assume that the linear operator L is the infinitesimal generator of a strongly
continuous semigroup {S(t)}t≥0 on X.
I A strong solution on [0, t0) of this ACP is a function u : [0, t0) → X such
that
• u is continuous on [0, t0) and continuously differentiable on (0, t0);
• u(t) ∈ D(L) for 0 ≤ t < t0;
• u(t) satisfies the ACP for 0 ≤ t < t0.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Mild Solutions
I Suppose that u is a strong solution. Under suitable assumptions on N , u
will also satisfy the Duhamel equation
◦
Z
t
S(t − s)N (u(s)) ds,
u(t) = S(t) u +
0 ≤ t < t0.
0
I This leads to the following definition of a weaker-type of solution to the ACP.
Definition 3.3 A mild solution on [0, t0) of the semilinear ACP is a function
u : [0, t0) → X such that
(i) u is continuous on [0, t0)
(ii) u satisfies the Duhamel equation on [0, t0).
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Existence and Uniqueness Results
I N : X → X is said to satisfy a local Lipschitz conditionon X if, for each
◦
◦
◦
u ∈ X, there exists a closed ball Br (u) := {f ∈ X : kf − u k ≤ r} such
that
◦
kN (f ) − N (g)k ≤ kkf − gk, ∀f, g ∈ Br (u).
I N is Fréchet differentiable at f ∈ X if an operator Nf ∈ B(X) exists such
that
N (f + h) = N (f ) + Nf (h) + E(f, h),
where the remainder E satisfies
lim
khk→0
kE(f, h)k
khk
= 0.
I The operator Nf is the Fréchet derivative of N at f .
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Existence and Uniqueness Results
Theorem 3.5 Let L ∈ G(M, ω; X) and let N satisfy a local Lipschitz
condition on X.Then there exists a unique mild solution of the ACP on some
interval [0, tmax). Moreover, if tmax < ∞, then
ku(t)k → ∞ as t → t−
max .
Theorem 3.6 Let L ∈ G(M, ω; X) and let N be continuously Fréchet
differentiable on X. Then the mild solution of the semilinear ACP, with
◦
u ∈ D(L), is a strong solution.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations