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Kinetic models in natural sciences
Jacek Banasiak1
School of Mathematics, Statistics and Computer Science, University of
KwaZulu-Natal, Durban, South Africa
[email protected]
1 Introduction
1.1 Preliminaries
In our terminology, a kinetic type equation describes an evolution of a population of objects depending on certain attributes from a certain space, subject
to a given set of conservation laws. One of the natural ways to describe such
a population is by providing the density of the objects with respect to the
attributes and investigate how it changes in time. The density, say u(x), is
either the number of elements with an attribute x (if the number of possible attributes is finite or countable), or a gives the number of elements with
attributes in a set A, according to the formula
∫
u(x)dµ,
(1)
A
if x is a continuous variable.
In many cases we are interested in tracking the total number of elements
of the population which, for a time t, is given by
∑
u(x, t),
(2)
x∈Ω
if Ω is countable, and
∫
u(x, t)dx,
(3)
Ω
if Ω is a continuum, where the set Ω is the space of attributes.
Kinetic equations for u usually are built in the following way. Let u(x, t)
be the density of a quantity Q with respect to the attribute(s) x from Ω at a
time t. Then the equation is obtained by balancing, for any subset A of the
space of attributes,
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Jacek Banasiak
1. the loss of Q from A due to the change of their attributes to due internal
interactions;
2. the gain of Q in A due to the changes of individuals’ attributes outside A
to the ones from A;
3. transport through A.
This results in the ‘Master Equation’
∂t u(x, t) = [Ku](x, t) := [T u](x, t) + [Au](x, t) + [Bu](x, t),
(4)
where A is the loss operator, B is the gain operator, while the T describes
transport phenomena. Equation (4) is supplemented with the initial state of
the system
◦
u(x, 0) = u (x), x ∈ Ω.
(5)
Only in exceptional circumstances can the problem (4), (5) be solved. Usually,
we have to prove the existence, uniqueness and other relevant properties of the
solution u without knowing its explicit form. There are various ways of doing
this. We shall follow the dynamical systems approach. Here, the evolution of
the system using a family of operators (G(t))t≥0 , parameterised by time, that
◦
map an initial state u of the system to all subsequent states in the evolution;
that is, the solutions are represented as
◦
u(t) = G(t) u .
(6)
The solutions, or the states of the system, belong to some appropriate state
space which is chosen partly due to its relevance to the problem but also
for the mathematical convenience. By no means is this choice unique: it is a
mathematical intervention into the model.
In the processes discussed in these lectures, an appropriately defined integral of the density over the space of attributes is the the total amount of
individuals, or the total mass of the system. Due to the conservation laws
used to construct the equation, this integral is constant, or changes in some
pre-defined way.
From this point of view it is natural to consider such processes as evolutions
of densities in the so-called L1 spaces; that is, in


∫


L1 (Ω, µ) = u; ∥u∥ = |u|dµ < +∞ ,


Ω
where µ is an appropriate measure corresponding to the process. Such a space
will be our state space; that is, the state of the system will be described by a
density with finite total mass.
However, we can try to control the process using some other gauge function. For instance, if we were interested in controlling the maximal concentration of the particles, a more proper choice would be to use the functional
Kinetic models in natural sciences
3
sup |u(x)|
x∈Ω
as the gauge function.
This approach leads in a natural way to a class of abstract spaces called
the Banach spaces.
Interlude – Banach spaces and linear operators
In what follows we shall restrict our attention to the state spaces which are
Banach spaces. To recall, a Banach space is a vector space X, equipped with
a finite gauge function ∥ · ∥, called norm, satisfying ∥x∥ = 0 if and only if
x = 0, ∥αx∥ = |α|∥x∥ for each scalar α and ∥x + y∥ ≤ ∥x∥ + ∥y∥, x, y ∈ X
and which is complete with respect to the convergence defined by the norm
(a space is complete if it contains limits of all Cauchy sequences).
Example 1. We introduce another type of Banach spaces which will be used
throughout the lectures: the Sobolev spaces. They are constructed on the basis
of L1 (Ω).
In general considerations, when dealing with partial derivatives of functions, often only the order of the derivative is important. In such cases, to
shorten calculations, we introduce the following notation. Let α = (α1 , . . . , αn ),
αi ∈ N0 , i = 1, . . . , n, be a multi-index and denote |α| = α1 + · · · + αn . Then,
for a given (locally integrable) function u we denote any generalized (distributional) derivative of u of order |α| by
∂αu =
∂ |α|
αn u.
1
∂xα
1 · · · ∂xn
The Sobolev spaces W1m (Ω) are defined as
W1m (Ω) := {u ∈ L1 (Ω); ∂ α u ∈ L1 (Ω), |α| ≤ m} .
In the same way, starting from the space Lp (Ω) of functions integrable with
power p, we can define Sobolev spaces Wpm (Ω), p ∈ [1, ∞[. For p = ∞ the corresponding space L∞ (Ω) is the space of functions which are bounded almost
m
(Ω) is the space in which all generalized derivatives
everywhere on Ω, and W∞
up to the order m have this property as well.
An object intimately related with a Banach space is a linear operator. A
(linear) operator from X to Y is a linear function A : D(A) → Y , where
D(A) is a linear subspace of X, called the domain of A. We use the notation
(A, D(A)) to denote the operator A with domain D(A). By L(X, Y ) we denote
the space of all bounded operators between X and Y ; that is, the operators
for which
∥A∥ := sup ∥Ax∥ = sup ∥Ax∥ < +∞.
(7)
∥x∥≤1
∥x∥=1
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Jacek Banasiak
The space L(X, X) is abbreviated as L(X). We further define the kernel
Ker A (or the null-space of A) by
Ker A = {x ∈ D(A); Ax = 0}
and the range Ran A of A by
Ran A = {y ∈ Y ; Ax = y for some x ∈ D(A)}
Furthermore, the graph of A is defined as the set {(x, y) ∈ X × Y ; x ∈
D(A), y = Ax}. We say that the operator A is closed if its graph is a closed
subspace of X × Y . Equivalently, A is closed if and only if for any sequence
(xn )n∈N ⊂ D(A), if lim xn = x in X and lim Axn = y in Y , then x ∈ D(A)
n→∞
n→∞
and y = Ax.
An operator A in X is closable if the closure of its graph is itself a graph
of an operator. In such a case the operator whose graph is G(A) is called the
closure of A and denoted by A.
Example 2. Consider the operator Af = f ′ in C([0, 1]) and L1 ([0, 1]). Then,
[6, Example 2.3], A is unbounded in both spaces, closed in C([0, 1]) and not
closed, but closable, in L1 ([0, 1]).
In this way, (4) can be written as the Cauchy problem for an ordinary
differential equation in an appropriate Banach space X: find R+ ∋ t → u(t) ∈
X such that
◦
∂t u = Ku, t > 0,
u(0) = u∈ X,
(8)
where K : D(K) → X is a realization of the expression K, defined on some
subset D(K) of the chosen state space X. It is clear that a minimum requirement for D(K) is that [Ku](·) ∈ X for u ∈ D(K). It is important to
remember that the expression K usually has multiple realizations and finding
an appropriate one, such that with (K, D(K)) the problem (8) is well posed
(often called the generator of the process) is a very difficult task.
We mention the so-called maximal realization of the expression K, Kmax
defined as the restriction of K to
D(Kmax ) = {u ∈ X; x → [Ku](x) ∈ X}.
The generator may be, or may be not, equal to Kmax . In the former case,
typically (4) is uniquely solvable in X.
1.2 The models
In this section we shall discuss the examples which will be discussed in the
course.
Kinetic models in natural sciences
5
1.3 Transport on networks
The first example does not exactly fit into (4). We consider a network with
some substance flowing along the edges and being redistributed in the nodes.
The process of redistribution of the flow is the loss-gain process governed by
the Kirchoff’s law (flow-in = flow-out) and thus is an example of a kinetic
process as defined above.
The network under consideration is represented by a simple directed graph
G = (V (G), E(G)) = ({v1 , . . . , vn }, {e1 , . . . , em }) with n vertices v1 , . . . , vn
and m edges (arcs), e1 , . . . , em . We suppose that G is connected but not
necessarily strongly connected. Each edge is normalized so as to be identified with [0, 1] with the head at 0 and the tail at 1. The outgoing incidence matrix, Φ− = (ϕ−
ij )1≤i≤n,1≤j≤m , and the incoming incidence matrix,
+
+
Φ = (ϕij )1≤i≤n,1≤j≤m , of this graph are defined, respectively as
ϕ−
ij =
{
{
ej
1 if vi →
0 otherwise.
ϕ+
ij =
ej
1 if → vi
0 otherwise.
If the vertex vi has more than one outgoing edge, we place a non negative
weight wij on the outgoing edge ej such that for this vertex vi ,
∑
wij = 1,
j∈Ei
where Ei is defined by saying that j ∈ Ei if the edge ej is outgoing from
vi . Naturally, wij = 1 if Ei = {j} and, to shorten notation, we adopt the
convention that wij = 1 for any j if Ei = ∅. Then the weighted outgoing
−
by replacing each nonzero ϕ−
incidence matrix, Φ−
w , is obtained from Φ
ij
entry by wij . If each vertex has an outgoing edge, then Φ−
w is row stochastic,
T
hence Φ− (Φ−
w ) = In (where the superscript T denotes the transpose). The
(weighted) adjacency matrix A = (aij )1≤i,j≤n of the graph is defined be taking
ek
aij = wjk if there is ek such that vj →
vi and 0 otherwise; that is, A =
+
− T
Φ (Φw ) . An important role is played by the line graph Q of G. To recall
Q = (V (Q), E(Q)) = (E(G), E(Q)), where
E(Q) = {uv; u, v ∈ E(G), the head of u coincides with the tail of v}
= {εj }1≤j≤k .
By B we denote the weighted adjacency matrix for the line graph; that is,
T +
B = (Φ−
w) Φ .
(9)
If there is an outgoing edge at each vertex then, from the definition of B, we
see that it is column stochastic. A vertex v will be called a source if there are
no incoming edges towards it and a sink if there are no edges outgoing from
it.
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Jacek Banasiak
Example 1.1. Consider the following graph. For
Φ− , Φ+ are given below.



00
1100000
1 0
0 0 1 1 0 0 0



+


Φ− = 
0 0 0 0 1 0 0, Φ = 0 1
0 0 0 0 0 1 0
0 0
00
0000001
while the adjacency matrix is given by

0000
1 0 0 0

1 1 0 1

0 0 0 0
0110
this graph, the matrices
0
0
0
0
1
0
0
1
0
0
0
0
0
0
1
0
0
1
0
0

0
0

0
,
1
0

0
0

0
.
1
0
We are interested in a flow on a closed network G. Then the standard
assumption is that the flow satisfies the Kirchoff law at the vertices
m
∑
j=1
ϕ−
ij cj uj (1, t) = wij
m
∑
ϕ+
ij cj uj (0, t),
t > 0, i ∈ 1, . . . , n,
j=1
which, in this context, is the conservation of mass law: the total inflow of mass
per unit time equals the total outflow at each node (vertex) of the network.
Let uj (x, t) be the density of particles at position x and at time t ≥ 0
flowing along edge ej for x ∈ [0, 1]. The particles on ej are assumed to move
with velocity cj > 0 which is constant for each j. We consider a generalized
Kirchoff’s law by allowing for decrease/amplification of the flow at the entrances and exits at each vertex. Then the flow in the system is described
by

∂ u (x, t)
= cj ∂x uj (x, t), x ∈ (0, 1), t ≥ 0,


 t j
uj (x, 0)
= fj (x),
(10)
m
∑

−

ϕ+
 ϕij ξj cj uj (1, t) = wij
ik (γk ck uk (0, t)),
k=1
where γj > 0 and ξj > 0 are the absorption/amplification coefficients at,
respectively, the head and the tail of the edge ej . If γj = ξj = 1 for all
j = 1, · · · , m, then the boundary conditions simply describe the Kirchoff law
at the vertices.
Kinetic models in natural sciences
7
Remark 1. We observe that the boundary condition in (10) takes a special
form if vi is either a sink or a source. If it is a sink, then Ei = ∅ and, by the
convention above,
0=
m
∑
ϕ+
ik (γk ck uk (0, t)),
t > 0,
(11)
k=1
and
ϕ−
ij ξj cj uj (1, t) = 0,
t > 0, j = 1, . . . , m,
(12)
if it is a source. Clearly the last condition is nontrivial only if j ∈ Ei as then
ϕ−
ij ̸= 0.
Let us denote C = diag(cj )1≤j≤m , K = diag(ξj )1≤j≤m and G = diag(γj )1≤j≤m .
We consider (10) as an abstract Cauchy problem
ut = Au,
u(0) = f ,
(13)
in X = (L1 ([0, 1]))m , where A is the realization of the expression A =
(cj ∂x )1≤j≤m on the domain
D(A) = {u ∈ (W11 ([0, 1]))m ; u satisfies the boundary conditions in (10)}.
(14)
It can be proved that
D(A) = {u ∈ (W11 ([0, 1]))m ; u(1) = K−1 C−1 BGCu(0)}.
(15)
1.4 Epidemiological models with age structure
The gain and loss part of the model are given by the SIRS system
S ′ = −Λ(I)S + δI,
I ′ = Λ(I)S − (δ + γ)I,
R′ = γI,
(16)
where S, I, R are, respectively, the number of susceptibles, infectives and recovered (with immunity) and γ, δ are recovery rates with and without immunity. For many diseases the rates of infection and recovery significantly
vary with age. Thus the vital dynamics of the population and the infection
mechanism interact to produce a nontrivial dynamics. To model it, we assume
that the total population in the absence of disease can be modelled by the
linear McKendrick model describing the evolution in time of the density of
the population with respect to age a ∈ [0, ω], ω < ∞, denoted by n(a, t).
The evolution is driven by the processes of death and birth with vital rates
µ(a) and β(a), respectively. Due to epidemics, we split the population into
susceptibles, infectives and recovered,
8
Jacek Banasiak
n(a, t) = s(a, t) + i(a, t) + r(a, t),
so that the scalar McKendrick equation for n splits, according to (16), into
the system
∂t s(a, t) + ∂a s(a, t) + µ(a)s(a, t) = −Λ(a, i(·, t))s(a, t) + δ(a)i(a, t),
∂t i(a, t) + ∂a i(a, t) + µ(a)i(a, t) = Λ(a, i(·, t))s(a, t) − (δ(a) + γ(a))i(a, t),
∂t r(a, t) + ∂a r(a, t) + µ(a)r(a, t) = γ(a)i(a, t),
(17)
where now the rates are age specific, see [21]. The function Λ is the infection
rate (or the force of infection). In the the intercohort model we use
∫ω
Λ(a, i(·, t)) =
K(a, a′ )i(a′ , t)da′ ,
(18)
0
where K is a nonnegative bounded function which accounts for the age dependence of the infections. For instance, for a typical childhood disease, K
should be large for small a, a′ and close to zero for large a or a′ (not necessarily
0, as usually adults can contract them). The extreme case, of an intracohort
infection, is given by
Λ(a, i(·, t)) = K0 (a)i(a, t).
(19)
System (17) is supplemented by the boundary conditions
∫ω
β(a)(s(a, t) + (1 − p)i(a, t) + (1 − q)r(a, t))da,
s(0, t) =
0
∫ω
i(0, t) = p
β(a)i(a, t)da,
0
∫ω
r(0, t) = q
β(a)r(a, t)da,
(20)
0
where p, q ∈ [0, 1] are the vertical transmission parameters of infectiveness
and immunity, respectively. Finally, we prescribe the initial conditions
◦
s(a, 0) =s (a),
◦
i(a, 0) = i (a),
◦
r(a, 0) =r (a).
(21)
Fragmentation-coagulation processes
They may seem very specific, but the range of applications is vast:
•
Chemical engineering: polymerization/depolimerization processes, with
possible mass loss through dissolution, chemical reactions, oxidation etc,
or mass growth due to the deposition of material on the clusters.
Kinetic models in natural sciences
•
•
•
9
Biology: Blood cells’ coagulation and splitting, animal grouping, phytoplankton at the level of aggregates.
Planetology: merging of planetesimals.
Aerosol research: coagulation of smoke, smog and dust particles, droplets
in clouds.
Also, they are most rewarding kinetic processes to study from analytical point
of view. In the case of pure fragmentation standard modelling process leads
to the following equation:
∫∞
∂t u(x, t) = −a(x)u(x, t) +
a(y)b(x|y)u(y, t)dy,
(22)
x
u is the density of particles of mass x, a is the fragmentation rate and b
describes the distribution of particle masses x spawned by the fragmentation
of a particle of mass y. Further
∫∞
M (t) =
xu(x, t)dx
(23)
0
is the total mass of the ensemble at time t. Local conservation principle requires
∫y
xb(x|y)dx = y,
(24)
0
with the expected number of particles produced by a particle of mass y is
given by
∫y
n0 (y) = b(x|y)dx.
0
Fragmentation can be supplemented by growth/decay, transport or diffusion processes, [6, 12, 13], but we will not discuss them here.
If we combine the fragmentation process with coagulation, we will get
∫∞
∂t u(x, t) = −a(x)u(x, t) +
a(y)b(x|y)u(y, t)dy
(25)
x
∫∞
−u(x, t)
k(x, y)u(y, t)dy +
0
1
2
∫x
k(x − y, y)u(x − y, t)u(y, t)dy.
0
The coagulation kernel k(x, y) represents the likelihood of a particle of size x
attaching itself to a particle of size y and, for a moment, we assume that it is
a symmetric nonnegative positive function.
10
Jacek Banasiak
Since the fragmentation and coagulation processes just rearrange the mass
distribution among the clusters, (23) implies that the natural space for to
analyse the fragmentation-coagulation processes is


∫∞


X1 = L1 (R+ , xdx) = u; ∥u∥1 = |u(x)|xdx < +∞ .


0
However, for technical reasons, for coagulation processes it is important to
control also the number of particles, or even higher moments of the density.
The best results are obtained in spaces


∫∞


X1,α = L1 (R+ , (1 + xα )dx) = u; ∥u∥0,α = |u|(1 + xα )dx < +∞ ,


0
α ≥ 1.
2 The tools
2.1 Basic positivity concepts
The common feature of the introduced models is that the solution originating
from a nonnegative density should stay nonnegative; that is, the solution
operator should be a ‘positive’ operator. Since we are talking about general
Banach spaces, we have to define what we mean by a nonnegative element
of a Banach space. Though in all cases discussed here our Banach space is
an L1 (Ω, µ) space, where the nonnegativity of a function f is understood
as f (x) ≥ 0 µ-almost everywhere, it is more convenient to work in a more
abstract setting.
Defining Order
In a given vector space X an order can be introduced either geometrically,
by defining the so-called positive cone (in other words, what it means to be a
positive element of X), or through the axiomatic definition:
Definition 1. Let X be an arbitrary set. A partial order (or simply, an
order) on X is a binary relation, denoted here by ‘ ≥’, which is reflexive,
transitive, and antisymmetric, that is,
(1) x ≥ x for each x ∈ X;
(2) x ≥ y and y ≥ x imply x = y for any x, y ∈ X;
(3) x ≥ y and y ≥ z imply x ≥ z for any x, y, z ∈ X.
Kinetic models in natural sciences
11
The supremum of a set is its least upper bound and the infimum is the
greatest lower bound. The supremum and infimum of a set need not exist. For
a two-point set {x, y} we write x ∧ y or inf{x, y} to denote its infimum and
x ∨ y or sup{x, y} to denote supremum.
We say that X is a lattice if every pair of elements (and so every finite
collection of them) has both supremum and infimum.
From now on, unless stated otherwise, any vector space X is real.
Definition 2. An ordered vector space is a vector space X equipped with partial order which is compatible with its vector structure in the sense that
(4) x ≥ y implies x + z ≥ y + z for all x, y, z ∈ X;
(5) x ≥ y implies αx ≥ αy for any x, y ∈ X and α ≥ 0.
The set X+ = {x ∈ X; x ≥ 0} is referred to as the positive cone of X.
If the ordered vector space X is also a lattice, then it is called a vector
lattice or a Riesz space.
For an element x in a Riesz space X we can define its positive and negative
part, and its absolute value, respectively, by
x+ = sup{x, 0},
x− = sup{−x, 0},
|x| = sup{x, −x},
which are called lattice operations. We have
x = x+ − x− ,
|x| = x+ + x− .
(26)
The absolute value has a number of useful properties that are reminiscent of
the properties of the scalar absolute value.
As the next step, we investigate the relation between the lattice structure
and the norm, when X is both a normed and an ordered vector space.
Definition 3. A norm on a vector lattice X is called a lattice norm if
|x| ≤ |y|
implies
∥x∥ ≤ ∥y∥.
(27)
A Riesz space X complete under the lattice norm is called a Banach lattice.
Property (27) gives the important identity:
∥x∥ = ∥|x|∥,
x ∈ X.
(28)
AM - and AL-spaces
Two important classes of Banach lattices that play a significant role later are
provided by the AL- and AM - spaces.
Definition 4. We say that a Banach lattice X is
(i) an AL-space if ∥x + y∥ = ∥x∥ + ∥y∥ for all x, y ∈ X+ ,
12
Jacek Banasiak
(ii) an AM-space if ∥x ∨ y∥ = max{∥x∥, ∥y∥} for all x, y ∈ X+ .
Example 3. Standard examples of AM -spaces are offered by the spaces C(Ω),
where Ω is either a bounded subset of Rn , or in general, a compact topological
space. Also the space L∞ (Ω) is an AM -space. On the other hand, most known
examples of AL-spaces are the spaces L1 (Ω). These examples exhaust all
(up to a lattice isometry) cases of AM - and AL-spaces. However, particular
representations of these spaces can be very different.
2.2 Positive Operators
Definition 5. A linear operator A from a Banach lattice X into a Banach
lattice Y is called positive, denoted by A ≥ 0, if Ax ≥ 0 for any x ≥ 0.
An operator A is positive if and only if |Ax| ≤ A|x|. This follows easily
from −|x| ≤ x ≤ |x| so, if A is positive, then −A|x| ≤ Ax ≤ A|x|. Conversely,
taking x ≥ 0, we obtain 0 ≤ |Ax| ≤ A|x| = Ax.
A frequently used property of positive operators is given in the following
theorem.
Theorem 1. If A is an everywhere defined positive operator from a Banach
lattice to a normed Riesz space, then A is bounded.
Proof. If A were not bounded, then we would have a sequence (xn )n∈N satisfying∑∥xn ∥ = 1 and ∥Axn ∥ ≥ n3 , n ∈ N. Because X is a Banach space,
∞
x := n=1 n−2 |xn | ∈ X. Because 0 ≤ |xn |/n2 ≤ x, we have ∞ > ∥Ax∥ ≥
∥A(|xn |/n2 )∥ ≥ ∥A(xn /n2 )∥ ≥ n for all n, which is a contradiction.
⊓
⊔
The norm of a positive operator can be evaluated by
∥A∥ =
sup
x≥0, ∥x∥≤1
∥Ax∥.
(29)
Indeed, since ∥A∥ = sup∥x∥≤1 ∥Ax∥ ≥ supx≥0,∥x∥≤1 ∥Ax∥, it is enough to
prove the opposite inequality. For each x with ∥x∥ ≤ 1 we have |x| = x+ +
x− ≥ 0 with ∥x∥ = ∥|x|∥ ≤ 1. On the other hand, A|x| ≥ |Ax|, hence
∥A|x|∥ ≥ ∥|Ax|∥ = ∥Ax∥. Thus sup∥x∥≤1 ∥Ax∥ ≤ supx≥0,∥x∥≤1 ∥Ax∥ and the
statement is proved.
As a consequence, we note that if
0 ≤ A ≤ B ⇒ ∥A∥ ≤ ∥B∥.
(30)
Moreover, it is worthwhile to emphasize that if there exists K such that
∥Ax∥ ≤ K∥x∥ for x ≥ 0, then this inequality holds for any x ∈ X. Indeed,
by (29) we have ∥A∥ ≤ K and using the definition of the operator norm, we
obtain the desired statement.
Kinetic models in natural sciences
13
2.3 Relation Between Order and Norm
There is a useful relation between the order, norm (absolute value) and the
convergence of sequences in R – any monotonic sequence which is bounded (in
absolute value), converges. One would like to have a similar result in Banach
lattices. It turns out to be not so easy.
Existence of an order in some set X allows us to introduce in a natural
way the notion of (order) convergence. Proper definitions of order convergence
require nets of elements but we do not need to go to such details.
For a non-increasing sequence (xn )n∈N we write xn ↓ x if inf{xn ; n ∈
N} = x. For a non-decreasing sequence (xn )n∈N the symbol xn ↑ x have an
analogous meaning. Then we say that an arbitrary sequence (xn )n∈N is order
convergent to x if it can be sandwiched between two monotonic sequences
o
converging to x. We write this as xn → x. One of the basic results is:
Proposition 1. Let X be a normed lattice. Then:
(1) The positive cone X+ is closed.
(2) If (xn )n∈N is nondecreasing and limn→∞ xn = x in the norm of X, then
x = sup{xn ; n ∈ N}.
Analogous statement holds for nonincreasing sequences.
In general, the converse of Proposition 1(2) is false; that is, we may
have xn ↑ x but (xn )n∈N does not converge in norm. Indeed, consider
xn = (1, 1, 1 . . . , 1, 0, 0, . . .) ∈ l∞ , where 1 occupies only the n first positions.
Clearly, supn∈N xn = x := (1, 1, . . . , 1, . . .) but ∥xn − x∥∞ = 1.
Such a converse holds in a special class of Banach lattices, called Banach
lattices with order continuous norm. There we have, in particular, that 0 ≤
xn↑ x and xn ≤ x for all n ∈ N if and only if (xn )n∈N is a Cauchy sequence,
[1, Theorem 12.9].
All Banach lattices Lp (Ω) with 1 ≤ p < ∞ have order continuous norms.
On the other hand, neither L∞ (Ω) nor spaces of continuous functions C(Ω̄)
(if Ω does not consist of isolated points) have order continuous norm.
The requirement that (xn )n∈N must be order dominated often is too restrictive. The spaces we are mostly concerned belong to a class which have a
stronger property.
Definition 6. We say that a Banach lattice X is a KB-space (Kantorovič–
Banach space) if every increasing norm bounded sequence of elements of X+
converges in norm in X.
We observe that if xn ↑ x, then ∥xn ∥ ≤ ∥x∥ for all n ∈ N and thus any
KB-space has order continuous norm. Hence, spaces which do not have order
continuous norm cannot be KB-spaces. This rules out the spaces of continuous
functions, l∞ and L∞ (Ω) from being KB-spaces.
14
Jacek Banasiak
Any reflexive Banach space is a KB-space, [6, Theorem 2.82]. That ALspaces (so, in particular, all L1 spaces) are also KB-spaces follows from the
following simple argument.
Theorem 2. Any AL-space is a KB-space.
Proof. If (xn )n∈N is an increasing and norm bounded sequence, then for 0 ≤
xn ≤ xm , we have
∥xm ∥ = ∥xm − xn ∥ + ∥xn ∥
as xm − xn ≥ 0 so that
∥xm − xn ∥ = ∥xm ∥ − ∥xn ∥ = |∥xm ∥ − ∥xn ∥| .
Because, by assumption, (∥xn ∥)n∈N is monotonic and bounded, and hence
convergent, we see that (xn )n∈N is a Cauchy sequence and thus converges. ⊓
⊔
2.4 Complexification
All our models refer to models, solutions of which must be real. Thus, our
problems should be posed in real Banach spaces. However, to take full advantage of the tools of functional analysis, such as the spectral theory, it is
worthwhile to extend our spaces to include also complex valued functions,
so that they become complex Banach spaces. While the algebraic and metric
structure of Banach spaces can be easily extended to the complex setting, the
extension of the order structure must be done with more care. This is done
by the procedure called complexification.
Definition 7. Let X be a real vector lattice. The complexification XC of X
is the set of pairs (x, y) ∈ X × X where, following the scalar convention, we
write (x, y) = x + iy. Vector operations are defined as in scalar case
x1 + iy1 + x2 + iy2 = x1 + x2 + i(y1 + y2 ),
(α + iβ)(x + iy) = αx − βy + i(βx + αy).
The partial order in XC is defined by
x0 + iy0 ≤ x1 + iy1
if and only if
x0 ≤ x1 and y0 = y1 .
(31)
Remark 2. Note, that from the definition, it follows that x ≥ 0 in XC is
equivalent to x ∈ X and x ≥ 0 in X. In particular, XC with partial order (31)
is not a lattice.
Example 4. Any positive linear operator A on XC is a real operator; that is,
A : X → X. In fact, let X ∋ x = x+ − x− . By definition, Ax+ ≥ 0 and
Ax− ≥ 0 so Ax+ , Ax− ∈ X and thus Ax = Ax+ − Ax− ∈ X.
Kinetic models in natural sciences
15
It is a more complicated task to introduce a norm on XC because standard
product norms, in general, fail to preserve the homogeneity of the norm.
Since XC is not a lattice, we cannot define the modulus of z = x+iy ∈ XC
in a usual way. However, following an equivalent definition of the modulus in
the scalar case, for x + iy ∈ XC we define
|x + iy| = sup {x cos θ + y sin θ}.
θ∈[0,2π]
It can be proved that this element exists.
Such a defined modulus has all standard properties of the scalar complex
modulus. Thus, one can define a norm on the complexification XC by
∥z∥c = ∥x + iy∥c = ∥|x + iy|∥.
(32)
Properties (a)–(c) and |x| ≤ |z|, |y| ≤ |z| imply
1
(∥x∥ + ∥y∥) ≤ ∥z∥c ≤ ∥x∥ + ∥y∥,
2
hence ∥ · ∥c is a norm on XC which is equivalent to the Euclidean norm on
X × X, denoted by ∥ · ∥C . As the norm ∥ · ∥ is a lattice norm on X, we have
∥z1 ∥c ≤ ∥z2 ∥c , whenever |z1 | ≤ |z2 |, and ∥ · ∥c becomes a lattice norm on XC .
Definition 8. A complex Banach lattice is an ordered complex Banach space
XC that arises as the complexification of a real Banach lattice X, according
to Definition 7, equipped with the norm (32).
Remember: a complex Banach lattice is not a Banach lattice!
Any linear operator A on X can be extended to XC according to the
formula
AC (x + iy) = Ax + iAy.
We observe that if A is a positive operator between real Banach lattices X
and Y then, for z = x + iy ∈ XC , we have
(Ax)cos θ + (Ay)sin θ = A(x cos θ + y sin θ) ≤ A|z|,
therefore |AC z| ≤ A|z|. Hence for positive operators
∥AC ∥c = ∥A∥.
(33)
There are examples, where ∥A∥ < ∥AC ∥c .
Note that the standard Lp (Ω) and C(Ω) norms are of the type (32). These
spaces have a nice property of preserving the operator norm even for operators
which are not necessarily positive, see [6, p. 63].
Remark 3. If for a linear operator A, we prove that it generates a semigroup
of say, contractions, in X, then this semigroup will be also a semigroup of
contractions on XC , hence, in particular, A is a dissipative operator in the
complex setting. Due to this observation we confine ourselves to real operators
in real spaces.
16
Jacek Banasiak
Series of Positive Elements in Banach Lattices
We note the following two results which are series counterparts of the dominated and monotone convergence theorems in Banach lattices.
Theorem 3. Let (xn (t))n∈N be family of nonnegative sequences in a Banach
lattice X, parameterized by a parameter t ∈ T ⊂ R, and let t0 ∈ T .
(i) If for each n ∈ N the function t → xn (t) is non-decreasing and
lim xn (t) = xn in norm, then
t↗t0
lim
∞
∑
xn (t) =
t↗t0 n=0
∞
∑
xn ,
(34)
n=0
irrespective of whether the right hand side exists in X or ∥
sup{∥
N
∑
∞
∑
xn ∥ :=
n=0
xn ∥; N ∈ N} = ∞. In the latter case the equality should be
n=0
understood as the norms of both sides being infinite.
(ii) If lim xn (t) = xn in norm for each n ∈ N and there exists (an )n∈N such
t→t0
that xn (t) ≤ an for any t ∈ T, n ∈ N with
∞
∑
∥an ∥ < ∞, then (34) holds
n=0
as well.
∑∞
Remark 4. Note that∑if X is a KB-space, then limt↗t0 n=0 xn (t) ∈ X im∞
plies convergence of n=0 xn . In fact, since xn ≥ 0 (by closedness of the pos∑N
∑∞
itive ∑
cone), N → n=0 xn is non-decreasing, and hence either∑ n=0 xn ∈ X,
∞
∞
or ∥ n=0 xn ∥ = ∞, and in the latter case we have ∥limt↗t0 n=0 xn (t)∥ =
∞.
2.5 First semigroups
As mentioned before, we are concerned with methods of finding solutions of
the Cauchy problem:
Definition 9. Given a complex or real Banach space and a linear operator A
with domain D(A) and range RanA contained in X and also given an element
◦
◦
u∈ X, find a function u(t) = u(t, u) such that
1. u(t) is continuous on [0, ∞[ and continuously differentiable on ]0, ∞[ in
the norm of X,
2. for each t > 0, u(t) ∈ D(A) and
∂t u(t) = Au(t),
t > 0,
(35)
3.
lim u(t) = u0
t→0+
in the norm of X.
(36)
Kinetic models in natural sciences
17
A function satisfying all conditions above is called the classical solution of
(35), (36). If u(t) ∈ D(A) (and thus u ∈ C 1 ([0, ∞[, X)), then such a function
is called a strict solution to (35), (36).
To shorten notation, we denote by C k (I, X) a space of functions which,
for each t ∈ I ⊂ R satisfy u(t) ∈ X and are continuously differentiable k times
in t with respect to the norm of X. Thus, e.g. a classical solution u satisfies
u ∈ C([0, ∞[, X) ∩ C 1 (]0, ∞[, X).
Definitions and basic properties
If the solution to (35), (36) is unique, then we can introduce the family of
operators (G(t))t≥0 such that u(t, u0 ) = G(t)u0 . Ideally, G(t) should be defined on the whole space for each t > 0, and the function t → G(t)u0 should
be continuous for each u0 ∈ X, leading to well-posedness of (35), (36). Moreover, uniqueness and linearity of A imply that G(t) are linear operators. A
fine-tuning of these requirements leads to the following definition.
Definition 10. A family (G(t))t≥0 of bounded linear operators on X is called
a C0 -semigroup, or a strongly continuous semigroup, if
(i) G(0) = I;
(ii) G(t + s) = G(t)G(s) for all t, s ≥ 0;
(iii) limt→0+ G(t)x = x for any x ∈ X.
A linear operator A is called the (infinitesimal) generator of (G(t))t≥0 if
Ax = lim+
h→0
G(h)x − x
,
h
(37)
with D(A) defined as the set of all x ∈ X for which this limit exists. Typically
the semigroup generated by A is denoted by (GA (t))t≥0 .
If (G(t))t≥0 is a C0 -semigroup, then the local boundedness and (ii) lead
to the existence of constants M > 0 and ω such that for all t ≥ 0
∥G(t)∥X ≤ M eωt
(38)
(see, e.g., [30, p. 4]). We say that A ∈ G(M, ω) if it generates (G(t))t≥0
satisfying (38). The type, or uniform growth bound, ω0 (G) of (G(t))t≥0 is
defined as
ω0 (G) = inf{ω; there is M such that (38) holds}.
(39)
Let (G(t))t≥0 be the semigroup generated by the operator A. From (37) and
conditions (ii), (iii) of Definition 10, we see that if A is the generator of
◦
◦
(G(t))t≥0 , then for u∈ D(A) the function t → G(t) u is a classical solution of
the following Cauchy problem,
18
Jacek Banasiak
∂t u(t) = Au(t),
t > 0,
◦
lim u(t) = u .
t→0+
(40)
(41)
We note that ideally the generator A should coincide with A but in reality
very often it is not so. However, for most of this chapter we are concerned with
solvability of (40), (41); that is, with the case when A of (35) is the generator
of a semigroup.
◦
We emphasize that, in general, the function u(t) = G(t) u is a classical so◦
◦
lution to (40), (41) only for u∈ D(A). For u∈ X \ D(A), however, the function
◦
u(t) = G(t) u is continuous but, in general, not differentiable. Nevertheless,
∫t
it follows that v(t) = 0 u(s)ds ∈ D(A) and u satisfies the integrated version
of (40), (41):
∫t
◦
u(t) = A u(s)ds+ u .
(42)
0
We say that a function u satisfying (??) (or, equivalently, (42)) is a mild
solution or integral solution of (40), (41).
Thus, if we have a semigroup, we can identify the Cauchy problem of which
it is a solution. Usually, however, we are interested in the reverse question,
that is, in finding the semigroup for a given equation. The answer is given
by the Hille–Yoshida theorem (or, more properly, the Feller–Miyadera–Hille–
Phillips–Yosida theorem). Before, however, we need to recall some terminology
related to the spectrum of an operator.
Interlude - the spectrum of an operator
Let us recall that the resolvent set of A is defined by
ρ(A) = {λ ∈ C; (λ − A)−1 ∈ L(X)}
and, for λ ∈ ρ(A), we define the resolvent of A by
R(λ, A) = (λI − A)−1 .
The complement of ρ(A) in C is called the spectrum of A and denoted
by σ(A). In general, it is possible that either ρ(A) or σ(A) is empty. The
spectrum is usually subdivided into several subsets.
•
•
•
Point spectrum σp (A) is the set of λ ∈ σ(A) for which the operator λI − A
is not one-to-one. In other words, σp (A) is the set of all eigenvalues of A.
Residual spectrum σr (A) is the set of λ ∈ σ(A) for which λI − A is oneto-one but Ran (λI − A) is not dense in X.
Continuous spectrum σc (A) which is the set of λ ∈ σ(A) for which the
operator λI − A is one-to-one and its range is dense in, but not equal to,
X
Kinetic models in natural sciences
19
The resolvent of any operator A satisfies the resolvent identity
R(λ, A) − R(µ, A) = (µ − λ)R(λ, A)R(µ, A),
λ, µ ∈ ρ(A).
(43)
For any bounded operator, the spectrum is a compact subset of C so that
ρ(A) ̸= ∅. If A is bounded, then the limit
√
r(A) = lim n ∥An ∥
(44)
n→∞
exists and is called the spectral radius. Clearly, r(A) ≤ ∥A∥. Equivalently,
r(A) = sup |λ|.
(45)
λ∈σ(A)
For an unbounded operator A the role of the spectral radius often is played
by the spectral bound s(A) defined as
s(A) = sup{ℜλ; λ ∈ σ(A)}.
(46)
Hille-Yosida theorem
Theorem 4. A ∈ G(M, ω) if and only if
(a) A is closed and densely defined,
(b) there exist M > 0, ω ∈ R such that (ω, ∞) ⊂ ρ(A) and for all
n ≥ 1, λ > ω,
M
∥(λI − A)−n ∥ ≤
.
(47)
(λ − ω)n
If A is the generator of (G(t))t≥0 then, for any x ∈ X, ℜλ > ω
∫∞
R(λ, A)x =
e−λt G(t)xdt.
(48)
0
A widely used approximation formula, which can also be used in the generation proof, is the operator version of the well-known scalar formula: if A is
the generator of a C0 -semigroup (G(t))t≥0 , then for any x ∈ X,
(
)−n
( n ( n ))n
t
I− A
x = lim
R
,A
x
n→∞
n→∞ t
n
t
G(t)x = lim
and the limit is uniform in t on bounded intervals, [30, Theorem 1.8.3].
(49)
20
Jacek Banasiak
Dissipative operators and contractive semigroups
Let X be a Banach space (real or complex) and X ∗ be its dual. From the
Hahn–Banach theorem, for every u ∈ X there exists u∗ ∈ X ∗ satisfying
<u∗ , u>= ∥u∥2 = ∥u∗ ∥2 .
Therefore the duality set
J (u) = {u∗ ∈ X ∗ ; <u∗ , u>= ∥u∥2 = ∥u∗ ∥2 }
(50)
is nonempty for every u ∈ X.
Definition 11. We say that an operator (A, D(A)) is dissipative if for every
u ∈ D(A) there is u∗ ∈ J (u) such that
ℜ <u∗ , Ax> ≤ 0.
(51)
An important equivalent characterisation of dissipative operators, [30,
Theorem 1.4.2], is that A is dissipative if and only if for all λ > 0 and
u ∈ D(A),
∥(λI − A)u∥ ≥ λ∥u∥.
(52)
Combination of the Hille–Yosida theorem with the above property gives a
generation theorem for dissipative operators, known as the Lumer–Phillips
theorem ([30, Theorem 1.43] or [20, Theorem II.3.15]).
Theorem 5. For a densely defined dissipative operator (A, D(A)) on a Banach space X, the following statements are equivalent.
(a) The closure A generates a semigroup of contractions.
(b) Ran(λI − A) = X for some (and hence all) λ > 0.
If either condition is satisfied, then A satisfies (51) for any u∗ ∈ J (u).
In other words, to prove that (the closure of) a dissipative operators generates a semigroup, we only need to show that the equation
λu − Au = f
(53)
is solvable for f from (a dense subset of) X for some λ > 0. We do not need
to prove that the solution is a resolvent satisfying (47).
In particular, if we know that A is closed, then the density of Ran(λI − A)
is sufficient for A to be a generator. On the other hand, if we do not know
a priori that A is closed, then Ran(λI − A) = X yields A being closed and
consequently that it is the generator.
Example 5. Let us have a look at the classical problem which often is incorrectly solved. Consider
Au = −∂x u,
x ∈ (0, 1),
Kinetic models in natural sciences
21
on D(A) = {u ∈ W11 (I); u(0) = 0}, where I =]0, 1[. The state space is real
X = L1 (I). For a given u ∈ X, we have
{
∥u∥signu(x)
if u(x) ̸= 0,
J (u) =
α ∈ [−∥u∥, ∥u∥] if u(x) = 0.
Note that J is a multivalued function. Further, by, say, [15], any element of
W11 (I) can be represented by an absolutely continuous (and thus continuous)
function on I.
Now, for v ∈ J (u) we have
∫1
<−∂x u, v> = −∥u∥
∂x u(x)signu(x)dx
0

∫

= −∥u∥ 
∫
∂x u(x)dx −
{x∈I; u(x)>0}


∂x u(x)dx .
{x∈I; u(x)<0}
Since u is continuous, both sets I+ := {x ∈ I; u(x) > 0} and I− := {x ∈
I; u(x) < 0} are open. Then, see [2, p. 42],
∑
I± =
]αn± , βn± [
n
where
Then
]αn± , βn± [
are non overlapping open intervals and the sum is countable.
∫
∂x u(x)dx =
∑
(u(βn± ) − u(αn± )) =
{
n
I±
u(1) if 1 ∈ I±
0
if 1 ∈
/ I±
as 1 only can be the right end of the component intervals and we used u(0) = 0.
Now, if 1 ∈ I+ , then u(1) > 0, if 1 ∈ I− , then u(1) < 0, and if 1 ∈
/ I+ ∪ I− ,
then u(1) = 0. In any case,
<−∂x u, v>≤ 0
and the operator (A, D(A)) is dissipative. Clearly, the solution of
λu + ∂x u = f,
is given by u(x) = e−λx
∫x
eλs f (s)ds and, for λ > 0,
0
∫1
∥u∥ ≤
0
u(0) = 0,

e−λx 
∫x

eλs |f (s)|ds dx ≤ ∥f ∥
0
which gives solvability of (53). We note that, of course, with a more careful
integration we would be able to obtain the Hille-Yosida estimate (47). This
additional work is, however, not necessary for dissipative operators.
22
Jacek Banasiak
Analytic semigroups
In the previous paragraph we noted that if an operator is dissipative, then we
can prove that it generates a semigroup provided (53) is solvable. It turns out
that the solvability of (53) can be used to prove that A generates a semigroup
without assuming that it is dissipative, but then we must consider complex λ.
Hence, let the inverse (λI − A)−1 exists in the sector
π
S π2 +δ := {λ ∈ C; |arg λ| < + δ} ∪ {0},
(54)
2
for some 0 < δ < π2 , and let there exist C such that for every 0 ̸= λ ∈ S π2 +δ
the following estimate holds:
∥R(λ, A)∥ ≤
C
.
|λ|
(55)
Then A is the generator of a uniformly bounded semigroup (GA (t))t≥0 (the
constant M in (38) not necessarily equals C) and (A(t))t≥0 is given by the
integral
∫
1
GA (t) =
eλt R(λ, A)dλ,
(56)
2πi
Γ
where Γ is an unbounded smooth curve in S π2 +δ . The reason why (A(t))t≥0
is called an analytic semigroup is that it extends to an analytic function on
Sδ .
The estimate (55) is sometimes awkward to prove as it requires the knowledge of the resolvent in the whole sector. The result given in [20, Theorem II
4.6] allows for restriction of the estimates to a positive half plane.
Theorem 6. An operator (A, D(A)) on a Banach space X generates a bounded
analytic semigroup (GA (z))z∈Sδ in a sector Sδ if and only if A generates a
bounded strongly continuous semigroup (GA (t))t≥0 and there exists a constant
C > 0 such that
C
∥R(r + is, A)∥ ≤
(57)
|s|
for all r > 0 and 0 ̸= s ∈ R.
This result can be generalized to arbitrary analytic semigroups: (A, D(A))
generates an analytic semigroup (GA (z))z∈Sδ if and only if A generates a
strongly continuous semigroup (GA (t))t≥0 and there exist constants C >
0, ω > 0 such that
C
∥R(r + is, A)∥ ≤
(58)
|s|
for all r > ω and 0 ̸= s ∈ R.
If A is the generator of an analytic semigroup (GA (t))t≥0 , then t → GA (t)
◦
has derivatives of arbitrary order on ]0, ∞[. This shows that t → GA (t) u
◦
solves the Cauchy problem (36) for arbitrary u ∈ X. This is a significant
◦
improvement upon the case of C0 -semigroup, for which u ∈ D(A) was required.
Kinetic models in natural sciences
23
Fractional powers of generators and interpolation spaces
If A generates an analytic semigroup, the formula (56) can be regarded as
the extension of the definition of etA via the so-called Dunford integral type
functional calculus, [30]. In a similar way we can define f (A) where f is any
scalar function which is analytic in an open neighbourhood of the spectrum
of A and such that the integral (56) is convergent.
One of the most important choices is
f (λ) = λ−α ,
where λ−α is real for λ > 0 and has a cut along the negative real axis. This
gives rise to bounded operators (−A)−α defined by
∫
1
(−A)−α =
λ−α (A − λI)−1 dλ,
(59)
2πi
Γ
The change of sign is dictated by the fact that changing by A to (−A), we
also change the position of the spectrum with respect to the integration curve.
Provided 0 ∈
/ ρ(−A), by inversion, we define unbounded operators (−A)α .
We denote by D((−A)α ) the domain of (−A)α . It follows that
D(A) ⊂ D((−A)α ) ⊂ X
if 0 < α < 1. For example, if A = ∆ on the maximal domain in L2 (Rn ), then
D(A) = W22 (Rn ) and D((−A)α ) = W22α (Rn ).
We note an important property of fractional powers of generators and of
corresponding analytic semigroups, which will used in the sequel. If (A(t))t≥0
is an analytic semigroup, then for every t > 0 and α ≥ 0, we have
(−A)α GA (t) = GA (t)(−A)α , the operator (−A)α GA (t) is bounded and
∥tα (−A)α GA (t)∥ ≤ Mα
(60)
for some constant Mα .
Example 6. Consider the simple example which will be useful later. Let X =
L1 ([0, ∞[) and [Au] = −a(x)u(x) where a(x) ≥ a0 > 0. Then −Au = au and
ρ(−A) = {λ ; λ ∈
/ Ran a}. Using the definition, we find
∫
1
λ−α
(−A)−α =
dλ,
2πi
a(x) − λ
Γ
where x is a parameter. The integral can be evaluated by closing the contour
by an arc running in the positive half-plane and enclosing the only pole of
the integrand at λ = a(x). Thus, noting the now the contour orientation is
clockwise and using the Cauchy theorem, we find
[(−A)−α u](x) = a−α (x)u(x).
24
Jacek Banasiak
The spaces D((−A)α ) serve a an important class of intermediate spaces
between D(A) and X. However, in some situations they are not sufficient.
Formula (60) can be written as
∥t1−α (−A)1−α GA (t)(−A)α x∥ ∈ L∞ (]0, 1[)
whenever x ∈ D((−A)α ) which could serve as a characterization of D((−A)α ).
Taking this as a starting point, let (A, D(A)) be the generator of an analytic semigroup (GA (t))t≥0 on a Banach space X. Then we construct a family
of intermediate spaces, DA (α, r), 0 < α < 1, 1 ≤ r ≤ ∞ in the following way:
DA (α, r) := {x ∈ X : t → v(t) := ∥t1−α−1/r AGA (t)x∥X ∈ Lr (I)}, (61)
∥x∥DA (α,r) := ∥x∥X + ∥v(t)∥Lr (I) ,
(62)
where I := (0, 1); see [27, p.45]. It turns out that these spaces can be identified
with real interpolation spaces between X and D(A) and one can use a rich
theory of the latter. In particular, by [27, Corollary 2.2.3], these spaces do
not depend explicitly on A, but only on D(A) and its graph norm. This is in
contrast to D((−A)α ) which only satisfy
DA (α, 1) ⊂ D((−A)α ) ⊂ DA (α, ∞)
(63)
and may depend on the particular form of the operator A.
Which makes the spaces DA (α, r) as useful as D((−A)α ) in dealing with
the semigroup generated by A is the fact that

∥R(λ, A)x∥DA (α,r) = ∥R(λ, A)x∥X + 

= ∥R(λ, A)x∥L(X) + 

∫1
1/r
∥s1−α−1/r AGA (s)R(λ, A)x∥rX 
0
1/r
∫1
∥R(λ, A)(s
1−α−1/r
0


≤ ∥R(λ, A)∥L(X) ∥x∥X + 
∫1
AGA (s)x)∥rX ds
1/r 

∥(s1−α−1/r AGA (s)x)∥rX ds 
0
≤ ∥R(λ, A)∥L(X) ∥x∥DA (α,r) .
This leads to the following observation.
Proposition 2. Let Aα,r be the part of A in DA (α, r). Then ρ(Aα,r ) ⊂ ρ(A),
∥R(λ, Aα,r ∥L(DA (α,r)) ≤ ∥R(λ, A)∥L(X) for λ ∈ ρ(A). Consequently, Aα,r generates an analytic semigroup in DA (α, r).
Kinetic models in natural sciences
25
Nonhomogeneous Problems
Let us consider the problem of finding the solution to the Cauchy problem:
∂t u = Au + f (t),
0 < t < T,
◦
u(0) = u,
(64)
where 0 < T ≤ ∞, A is the generator of a semigroup and f : (0, T ) → X is a
known function.
If we are interested in classical solutions, then clearly f must be continuous. However, this condition proves to be insufficient. We thus generalise
the concept of the mild solution introduced in (42). We observe that if u is a
classical solution of (64), then it must be given by
◦
∫t
G(t − s)f (s)ds
u(t) = G(t) u +
(65)
0
(see, e.g., [30, Corollary 4.2.2]). The integral is well defined even if f ∈
◦
L1 ([0, T ], X) and u∈ X. We call u defined by (65) the mild solution of (64).
For an integrable f such u is continuous but not necessarily differentiable, and
therefore it may be not a solution to (64).
We have the following theorem giving sufficient conditions for a mild solution to be a classical solution (see, e.g., [30, Corollary 4.2.5 and 4.2.6]).
Theorem 7. Let A be the generator of a C0 -semigroup (GA (t))t≥0 and x ∈
D(A). Then (65) is a classical solution of (64) if either
(i) f ∈ C 1 ([0, T ], X), or
(ii) f ∈ C([0, T ], X) ∩ L1 ([0, T ], D(A)).
If the semigroup (GA (t))t≥0 generated by A is analytic, then the requirements imposed on f can be substantially weakened. We have then the following
counterpart of Theorem 7.
Theorem 8. Let A be the generator of an analytic semigroup (GA (t))t≥0 ,
◦
u∈ X and f ∈ L1 ([0, T ], X). Then (65) is the classical solution of (64) if
either
(i) f is locally Hölder continuous on ]0, T [, or
(ii) ∃{α > 0} f ∈ C(]0, T ], X)∩L1 ([0, T ], D((−A)α )) and t → ∥(−A)α f (t)∥X
is bounded over compact subsets of ]0, T ].
Part (ii) of this theorem has been proved in [9].
An important refinement of Theorem 8 which becomes very useful in nonlinear problems is that, actually, the solution has a better regularity. In fact,
under assumption (i) we additionally have
0,1−r
u ∈ Cloc
(]0, T [, D((−A)r )),
0 ≤ r < 1.
(66)
By (63), the statement of the above theorem holds if the domains of the
fractional power are replaced by an appropriate D(α, 1).
26
Jacek Banasiak
Positive Semigroups
Definition 12. Let X be a Banach lattice. We say that the semigroup (G(t))t≥0
on X is positive if for any x ∈ X+ and t ≥ 0,
G(t)x ≥ 0.
We say that an operator (A, D(A)) is resolvent positive if there is ω such that
(ω, ∞) ⊂ ρ(A) and R(λ, A) ≥ 0 for all λ > ω.
Remark 5. In this section, because we address several problems related to
spectral theory, we need complex Banach lattices. Let us recall, Definitions 7
and 8, that a complex Banach lattice is always a complexification XC of an
underlying real Banach lattice X. In particular, x ≥ 0 in XC if and only if
x ∈ X and x ≥ 0 in X.
It is easy to see that a strongly continuous semigroup is positive if and only
if its generator is resolvent positive. In fact, the positivity of the resolvent for
λ > ω follows from (48) and closedness of the positive cone; see Proposition 1.
Conversely, the latter with the exponential formula (49) shows that resolvent
positive generators generate positive semigroups.
There are several results on the resolvent of a positive semigroup which
we collect in the following theorem.
Theorem 9. [29, Theorem 1.4.1] Let (GA (t))t≥0 be a positive semigroup on
a Banach lattice, with generator A. Then
∫∞
R(λ, A)x =
e−λt GA (t)xdt
(67)
0
for all λ ∈ C with ℜλ > s(A). Furthermore,
(i) Either s(A) = −∞ or s(A) ∈ σ(A);
(ii) For a given λ ∈ ρ(A), we have R(λ, A) ≥ 0 if and only if λ > s(A);
(iii) For all ℜλ > s(A) and x ∈ X, we have |R(λ, A)x| ≤ R(ℜλ, A)|x|.
We conclude this section by briefly describing an approach of [3] which
has a number of interesting applications.
To fix attention, assume for the time being that ω < 0 (thus, in particular,
A is invertible and −A−1 = R(0, A)) and λ > 0. The resolvent identity
−A−1 = (λI − A)−1 + λ(λI − A)−1 (−A−1 ),
can be extended by induction to
−A−1 = R(λ, A) + λR(λ, A)2 + · · · + λn R(λ, A)n (−A−1 ).
Now, because all terms above are nonnegative, we obtain
(68)
Kinetic models in natural sciences
27
sup {λn ∥(λ − A)−n (−A−1 )∥X } = M < +∞.
n∈N,λ>ω
This is ‘almost’ the Hille–Yosida estimate and allows us to prove that the
Cauchy problem (40), (41) has a mild Lipschitz continuous solution for
◦
u∈ D(A2 ). If, in addition, A is densely defined, then this mild solution is
differentiable, and thus it is a strict solution (see, e.g., [4] and [5, pp. 191–
200]). These results are obtained by means of the integrated, or regularised,
semigroups, which are beyond the scope of this lecture, so we do not enter into
details of this very rich field. We mention, however, an interesting consequence
of (68) for semigroup generation which has already found several applications
and which we use later.
Theorem 10. [3, 14] Let A be a densely defined resolvent positive operator.
If there exist λ0 > s(A), c > 0 such that for all x ≥ 0,
∥R(λ0 , A)x∥X ≥ c∥x∥X ,
(69)
then A generates a positive semigroup (GA (t))t≥0 on X.
Proof. Let us take s(A) < ω ≤ λ0 and set B = A − ωI so that s(B) < 0.
Because R(0, B) = R(ω, A) ≥ R(λ0 , A), it follows from (69) and (30) that
∥R(0, B)x∥X ≥ ∥R(λ0 , A)x∥X ≥ c∥x∥X
for x ≥ 0. Using (68) for B and taking x = λn R(λ, B)n g, g ≥ 0, we obtain,
by (69),
∥λn R(λ, B)n g∥X ≤ c−1 ∥R(0, B)λn R(λ, B)n g∥ ≤ c−1 ∥R(0, B)g∥X ≤ M ∥g∥X ,
for λ > 0. Again, by (30), we can extend the above estimate onto X proving the
Hille–Yosida estimate. Because B is densely defined, it generates a bounded
positive semigroup and thus ∥GA (t)∥ ≤ M eωt .
⊓
⊔
Perturbation techniques
Verifying conditions of the Hille–Yosida, or even the Lumer–Phillips, theorems
for a concrete problem is quite often a formidable task. On the other hand,
in many cases the operator appearing in the evolution equation at hand is
built as a combination of much simpler operators that are relatively easy to
analyse. The question now is to what extent the properties of these simpler
operators are inherited by the full equation. More precisely, we are interested
in the problem:
Problem P. Let (A, D(A)) be a generator of a C0 -semigroup on a
Banach space X and (B, D(B)) be another operator in X. Under what
conditions does A + B, or an extension K of A + B, generate a C0 semigroup on X?
28
Jacek Banasiak
We note that the situation when K = A + B is quite rare. Usually at best we
can show that there is an extension of A+B (another realization of K = A+B)
which is the generator. The reason for this is that, unless B is in some sense
strictly subordinated to A, adding B to A may significantly alter some vital
properties of A. The identification of K in such cases usually is a formidable
task.
A Spectral Criterion
Usually the first step in establishing whether A + B, or some of its extensions,
generates a semigroup is to find if λI − (A + B) (or its extension) is invertible
for all sufficiently large λ.
In all cases discussed here we have the generator (A, D(A)) of a semigroup
and a perturbing operator (B, D(B)) with D(A) ⊆ D(B).
We note that B is A-bounded; that is, for some a, b ≥ 0 we have
∥Bx∥ ≤ a∥Ax∥ + b∥x∥,
x ∈ D(A)
(70)
if and only if BR(λ, A) ∈ L(X) for λ ∈ ρ(A).
In what follows we denote by K an extension of A + B. We now present an
elegant result relating the invertibility properties of λI − K to the properties
of 1 as an element of the spectrum of BR(λ, A), first derived in [?].
Theorem 11. Assume that Λ = ρ(A) ∩ ρ(K) ̸= ∅.
(a) 1 ∈
/ σp (BR(λ, A)) for any λ ∈ Λ;
(b) 1 ∈ ρ(BR(λ, A)) for some/all λ ∈ Λ if and only if D(K) = D(A) and
K = A + B;
(c) 1 ∈ σc (BR(λ, A)) for some/all λ ∈ Λ if and only if D(A)
K = A + B;
D(K) and
(d) 1 ∈ σr (BR(λ, A)) for some/all λ ∈ Λ if and only if K ) A + B.
Corollary 1. Under the assumptions of Theorem 11, K = A + B if one of
the following criteria is satisfied: for some λ ∈ ρ(A) either
(i) BR(λ, A) is compact (or, if X = L1 (Ω, dµ), weakly compact), or
(ii) the spectral radius r(BR(λ, A)) < 1.
Proof. If (ii) holds, then obviously I − BR(λ, A) is invertible by the Neumann series theorem:
(I − BR(λ, A))−1 =
∞
∑
(BR(λ, A))n ,
n=0
giving the thesis by Proposition 11 (b). Additionally, we obtain
(71)
Kinetic models in natural sciences
R(λ, A + B) = R(λ, A)(I − BR(λ, A))−1 = R(λ, A)
∞
∑
29
(BR(λ, A))n . (72)
n=0
If (i) holds, then either BR(λ, A) is compact or, in L1 setting, (BR(λ, A))2 is
compact, [19, p. 510], and therefore, if I − BR(λ, A) is not invertible, then 1
must be an eigenvalue, which is impossible by Theorem 11(c).
⊓
⊔
If we write the resolvent equation
(λI − (A + B))x = y,
y ∈ X,
(73)
in the (formally) equivalent form
x − R(λ, A)Bx = R(λ, A)y,
(74)
then we see that we can hope to recover x provided the Neumann series
R(λ)y :=
∞
∑
(R(λ, A)B)n R(λ, A)y =
n=0
∞
∑
R(λ, A)(BR(λ, A))n y.
(75)
n=0
is convergent. Clearly, if (71) converges, then we can factor out R(λ, A) from
the series above getting again (72). However, R(λ, A) inside acts as a regularising factor and (75) converges under weaker assumptions than (71) and this
fact is frequently used to construct the resolvent of an extension of A + B
(see, e.g., Theorem 15).
The most often used perturbation theorem is the Bounded Perturbation
Theorem and Related Results, see e.g. [20, Theorem III.1.3]
Theorem 12. Let (A, D(A)) ∈ G(M, ω) for some ω ∈ R, M ≥ 1. If B ∈
L(X), then (K, D(K)) = (A + B, D(A)) ∈ G(M, ω + M ∥B∥).
In many cases the Bounded Perturbation Theorem gives insufficient information. It can be combined with the Trotter product formula, [20, 30].
Assume K0 is of type (1, ω0 ), and K1 is of type (1, ω1 ), ω0 , ω1 ∈ R,. If we
know that (K, D(K0 ) ∩ D(K1 )) := (K0 + K1 , D(K0 ) ∩ D(K1 )) generates a
semigroup, then
n
GK (t)x = lim (GK0 (t/n)GK1 (t/n)) x,
n→∞
x ∈ X,
(76)
uniformly in t on compact intervals and K is of type (1, ω) with ω = ω0 + ω1 .
Moreover, if both semigroups (GK0 (t))t≥0 and (GK1 (t))t≥0 are positive, then
(GK (t))t≥0 is positive.
The assumption of boundedness of B, however, is often too restrictive.
Another frequently used result uses special structure of dissipative operators.
Theorem 13. Let A and B be linear operators in X with D(A) ⊆ D(B) and
A + tB is dissipative for all 0 ≤ t ≤ 1. If
∥Bx∥ ≤ a∥Ax∥ + b∥x∥,
(77)
for all x ∈ D(A) with 0 ≤ a < 1 and for some t0 ∈ [0, 1] the operator
(A+t0 B, D(A)) generates a semigroup (of contractions), then A+tB generates
a semigroup of contractions for every t ∈ [0, 1].
30
Jacek Banasiak
2.6 Positive perturbations of positive semigroups
Perturbation results can be significantly strengthened in the framework of
positive semigroups. We have seen in (71) that the condition r(BR(λ, A)) < 1
implies invertibility of λI − (A + B). It turns out that this condition is equivalent to invertibility for positive perturbations of resolvent positive operators.
Theorem 14. Assume that X is a Banach lattice. Let A be a resolvent positive operator in X and λ > s(A). Let B : D(A) → X be a positive operator.
Then the following are equivalent,
(a) r(B(λI − A)−1 ) < 1;
(b) λ ∈ ρ(A + B) and (λI − (A + B))−1 ≥ 0.
If either condition is satisfied, then
(λI − A − B)−1 = (λI − A)−1
∞
∑
(B(λI − A − B)−1 )n ≥ (λI − A)−1 . (78)
n=0
Kato-Voigt type results
Here we consider only X = L1 (Ω, dµ) where (Ω, µ) is a measure space. We recall that if Z ⊂ X is a subspace, then by Z+ we denote the cone of nonnegative
elements of Z and for f ∈ X, the symbols f± denote the positive and negative
part of f ; that is, f+ = max{f, 0} and f− = − min{f, 0}. Let (G(t))t≥0 be a
strongly continuous semigroup on X. We say that (G(t))t≥0 is a substochastic
semigroup if for any t ≥ 0 and x ≥ 0, G(t)x ≥ 0 and ∥G(t)x∥ ≤ ∥x∥, and a
stochastic semigroup if additionally ∥G(t)f ∥ = ∥f ∥ for f ∈ X+ .
Theorem 15. Let X = L1 (Ω) and suppose that the operators A and B satisfy
1. (A, D(A)) generates a substochastic semigroup (GA (t))t≥0 ;
2. D(B) ⊃ D(A) and Bu ≥ 0 for u ∈ D(B)+ ;
3. for all u ∈ D(A)+
∫
(Au + Bu)dµ ≤ 0.
(79)
Ω
Then there is an extension (K, D(K)) of (A + B, D(A)) generating a C0 semigroup of contractions, say, (GK (t))t≥0 . The generator K satisfies
R(λ, K)u =
∞
∑
R(λ, A)(BR(λ, A))k u,
k=0
λ > 0.
(80)
Kinetic models in natural sciences
31
Proof. First, assumption (79) gives us dissipativity on the positive cone. Next,
let us take u = R(λ, A)x = (λI − A)−1 x for x ∈ X+ so that u ∈ D(A)+ .
Because R(λ, A) is a surjection from X onto D(A), by
(A + B)u = (A + B)R(λ, A)x = −x + BR(λ, A)x + λR(λ, A)x,
we have
∫
−
∫
x dµ +
Ω
∫
R(λ, A)x dµ ≤ 0.
BR(λ, A)x dµ + λ
Ω
(81)
Ω
Rewriting the above in terms of the norms, we obtain
λ∥R(λ, A)x∥ + ∥BR(λ, A)x∥ − ∥x∥ ≤ 0,
x ∈ X+ ,
(82)
from which ∥BR(λ, A)∥ ≤ 1.
We define operators Kr , 0 ≤ r < 1 by Kr = A + rB, D(Kr ) = D(A). We
see that the spectral radius of rBR(λ, A) does not exceed r < 1, the resolvent
(λI − (A + rB))−1 exists and is given by
R(λ, Kr ) := (λI − (A + rB))−1 = R(λ, A)
∞
∑
n
rn (BR(λ, A)) ,
(83)
n=0
where the series converges absolutely and each term is positive. Let x ∈
D(A)+ . Then we have, for r < 1,
∫
∫
(A + B)xdµ + (r − 1)
(A + rB)xdµ, =
Ω
∫
Ω
Bxdµ ≤ 0
(84)
Ω
by assumptions. Thus, by the above,
∫
∥(λI − Kr )x∥ ≥
∫
(λI − Kr )xdµ = λ∥x∥ −
Ω
Kr xdµ ≥ λ∥x∥,
Ω
for all x ∈ D(A)+ , by (84). Hence,
∥R(λ, Kr )y∥ ≤ λ−1 ∥y∥
(85)
for all y ∈ X+ and, because R(λ, Kr ) is positive, (85) can be extended to the
whole space X, by (29). Therefore, by the Lumer–Phillips theorem, for each
0 ≤ r < 1, (Kr , D(A)) generates a contraction semigroup which we denote
(Gr (t))t≥0 . Since (R(λ, Kr )x)0≤r<1 is increasing as r ↑ 1 for each x ∈ X+ ,
and {∥R(λ, Kr )x∥}0≤r<1 is bounded and X = L1 (Ω) is a KB-space, there is
an element yλ,x ∈ X+ such that
lim R(λ, Kr )x = yλ,x
r→1−
32
Jacek Banasiak
in X. By the Banach–Steinhaus theorem we obtain the existence of a bounded
positive operator on X, denoted by R(λ), such that R(λ)x = yλ,x . We use
the Trotter–Kato theorem to obtain that R(λ) is defined for all λ > 0 and
it is the resolvent of a densely defined closed operator K which generates a
semigroup of contractions (GK (t))t≥0 . Moreover, for any x ∈ X,
lim Gr (t)x = GK (t)x,
(86)
r→1−
and the limit is uniform in t on bounded intervals and, provided x ≥ 0,
monotone as r ↑ 1. By the monotone convergence theorem, Theorem 3,
R(λ, K)x =
∞
∑
R(λ, A)(BR(λ, A))k x,
x∈X
(87)
k=0
and we can prove that R(λ, K)(λI − (A + B))x = x which shows that K ⊇
A + B.
⊓
⊔
The identification of K is a much more difficult task.
We note that (79) can be written as
∫
(Au + Bu) dµ = −c(u) ≤ 0,
(88)
Ω
where c is a positive functional on D(T ). We assume that c has the monotone
convergence property: c(un ) → cu for un ↑ u, then cu = c(u) (for instance, it
is an integral functional). The following theorem is fundamental for characterizing the generator of the semigroup.
Theorem 16. For any fixed λ > 0, there is 0 ≤ βλ ∈ X ∗ with ∥βλ ∥ ≤ 1 such
that for any f ∈ X+ ,
λ∥R(λ, K)f ∥ = ∥f ∥− <βλ , f> − c (R(λ, K)f ) .
(89)
In particular, c extends to a nonnegative continuous linear functional on
D(K).
It turns out that
(BR(λ, T ))∗ βλ = βλ .
(90)
and hence, if βλ ̸= 0, then σp (BR(λ, T ))∗ ̸= ∅. This implies that σr (BR(λ, T )) ̸=
∅ and, by Theorem 11 d), K ̸= A + B.
Another result, though not as elegant, is often more useful. It is based on
the observation that the following are equivalent:
(a) K = T + B.
∫
(b) Ku dµ ≥ −c(u), u ∈ D(K)+ .
Ω
Though the implication (b) ⇒ (a) seems to be useless as it require the knowledge of K which is what we are looking for, we note that if we can prove it
for an extension of K, then obviously it will be valid for K. Hence
Kinetic models in natural sciences
Theorem 17. [6] If there exists an extension K such that
for all u ∈ D(K)+ , then K = A + B.
∫
Ω
33
Ku dµ ≥ −c(u)
Often one can use Kmax as the required extension of K.
Arendt-Rhandi theorem
Theorem 18. Assume that X is a Banach lattice, (A, D(A)) is a resolvent
positive operator which generates an analytic semigroup and (B, D(A)) is a
positive operator. If (λ0 I − (A + B), D(A)) has a nonnegative inverse for some
λ0 larger than the spectral bound s(A) of A, then (A + B, D(A)) generates a
positive analytic semigroup.
Proof. The proof is an application of Theorem 14. Under assumptions of
this theorem, we obtain that r(BR(λ0 , A)) < 1. In particular, the series
∞
∑
(BR(λ0 , A))n converges in the uniform operator topology. Next, by Then=0
orem 9, R(λ, A) ≥ 0 if and only if ρ(A) ∋ λ > s(A). Thus, using the resolvent
identity we have
R(λ, A) = R(λ0 , A) − (λ − λ0 )R(λ0 , A)R(λ, A) ≤ R(λ0 , A)
whenever λ ≥ λ0 . Since BR(λ, A) is bounded in X, see Theorem 1, B :
D(A) → X is bounded in the graph norm of D(A). Let us now take λ ∈ C with
ℜλ ≥ λ0 , R ∋ µ > λ0 and f ∈ D(A). Then µR(µ, A)R(λ, A)f → R(λ, A)f as
µ → ∞ in the graph norm of D(A), see e.g., [30, Lemmas 1.3.2 and 1.3.3] and
we have, for f ∈ D(A)
|BR(λ, A)f | = lim |BR(µ, A)R(λ, A)| ≤ lim BR(µ, A)R(ℜλ, A)|f |
µ→∞
µ→∞
= BR(µ, A)R(ℜλ, A)|f |,
where we used |R(λ, A)f | ≤ R(ℜλ, A)|f | for ℜλ > s(A), see Theorem 9. Thus,
by density,
|BR(λ, A)f | ≤ BR(ℜλ, A)|f |
(91)
for all f ∈ X and therefore
r(BR(λ, A)) ≤ r(BR(λ0 , A)) < 1
for any λ ∈ C with ℜλ ≥ λ0 . In particular,
∞
∑
(BR(λ, A))n converges to a
n=0
bounded linear operator with
∞
∞
∑
∑
n n
≤ Mλ0 ∥f ∥,
(BR(λ,
A))
f
≤
(BR(λ
,
A))
|f
|
0
n=0
n=0
uniformly for λ ∈ C with ℜλ > λ0 .
34
Jacek Banasiak
Next we consider analyticity issue. Using Theorem 6 for the operator A,
there are ωA and MA such that
∥R(r + is, A)∥ ≤
MA
|s|
for r > ωA . Taking now ω > max{λ0 , ωA } we have, by (78),
∞
∞
MA ∑
∑
n n ∥R(r + is, A + B)f ∥ = R(λ, A)
(BR(λ, A)) f ≤
(BR(λ, A)) f |s|
n=0
n=0
MA Mλ0
∥f ∥,
f ∈ X,
≤
|s|
for all r > ω. Therefore (A + B, D(A)) generates an analytic semigroup.
⊓
⊔
2.7 Semi-linear problems
Next we consider the semilinear abstract Cauchy problem
∂t u = Au + f (t, u),
t > 0,
◦
u(0) = u,
(92)
where A is a generator of a C0 -semigroup (GA (t))t≥0 and f : [0, T ] × X → X
is a known function. Since a priori we don’t know properties of the solution u
(which may even fail to exist), it is plausible to start from a weaker formulation
of the problem i.e. from the integral equation:
◦
∫t
GA (t − s)f (s, u(s))ds.
u(t) = GA (t) u +
(93)
0
This form is typical for fixed point techniques. We shall focus on the Banach
contraction principle which leads to Theorem 19 below. It requires a relatively
strong regularity from f as in the following definition.
We say that f : [0, T ] × X → X is locally Lipschitz continuous in u,
uniformly in t on bounded intervals if
∀{t′ ∈ [0, T [, c > 0}∃{L(c, t′ )}∀{t ∈ [0, t′ ], ∥u∥, ∥v∥ ≤ c}
such that
∥f (t, u) − f (t, u)∥X ≤ L(c, t′ )∥u − v∥X
Theorem 19. Let f : [0, ∞[×X → X be continuous in t, for t ∈ [0, ∞[ and
locally Lipschitz continuous in u, uniformly in t, on bounded intervals. If A
is the generator of a C0 -semigroup (GA (t))t≥0 on X then ∀u∈X
∃tmax >0 such
◦
that the problem (93) has a unique mild solution u on [0, tmax [ . Moreover, if
tmax < +∞, then lim ∥u(t)∥X = ∞.
t→∞
Kinetic models in natural sciences
35
The proof is done by Picard iterations as in the scalar case. Also, similarly
to the scalar case, a sufficient condition for the existence of a global mild
solution is that f be uniformly Lipschitz continuous on X. Uniform Lipschitz
continuity yields at most linear growth in ∥x∥ of ∥f (t, x)∥. In fact, even for
f (u) = u2 and A = 0, the blow-up occurs in finite time if the initial data are
sufficiently large.
There are two standard sufficient conditions ensuring that the mild solution, described in Theorem 19, is a classical solution. Both follow from the
corresponding results for nonhomogeneous problems. They are either that
f : [0, ∞[×X → X is continuously differentiable with respect to both variables, or that f : [0, ∞[×D(A) → D(A) is continuous. Certainly, in both
cases to ensure that the solution is a classical solution we must assume that
◦
u∈ D(A).
As in the subsection on nonhomogeneous problems, a substantial relaxation of the requirements can be achieved if A generates an analytic semigroup. A crucial role is played here by the domains of fractional powers of
generators. Let us denote Xα = D((−A)α ) with usual graph norm. We have
the following theorem.
Theorem 20. Let U ⊆ R × Xα , 0 < α < 1, be an open set and f : U → X
satisfy: ∀(t,x)∈U ∃(t,x)∈V ⊂U,L>0,0<θ≤1 ∀(ti ,xi )∈V
∥f (t1 , x1 ) − f (t2 , x2 )∥ ≤ L(|t1 − t2 |θ + ∥x1 − x2 ∥Xα ),
and let an invertible A, satisfying 0 ∈ ρ(A) be the generator of a bounded
◦
◦
analytic semigroup. If (0, u) ∈ U , then there is t1 = t1 (u) such that (92) has a
unique local classical solution u ∈ C([0, t1 [, X) ∩ C 1 (]0, t1 [, X). Moreover, the
solution continuously depends on the initial data and is not global in time if
either reaches the boundary U or its Xα norm blows up in finite time.
In fact, we have a better regularity result. If the constants θ and L are
uniform in U , then u ∈ C 1+ν (]0, t1 [, X); that is, ∂t u is Hölder continuous on
]0, t1 [ with ν = min{θ, β} with 0 < β < 1 − α.
We formulated the above theorem in the form usually found in the literature on dynamical systems. However, as was pointed out earlier, using
Xα = D((−A)α ) often is inconvenient as it may depend on the particular
form of A and there is no explicit method to evaluate the norm. However, in
[27, Chapter 7] we can find a parallel theory leading to the analogous theorem in which Xα can be any interpolation space discussed in the section on
analytic semigroups.
In particular, under assumptions of Theorem 20, in any interpolation space
Xα , the solution u is a strict solution; that is u ∈ u ∈ C([0, tmax [, D(A)) ∩
◦
◦
◦
C 1 ([0, t1 [, X), provided u∈ D(A) and A u +f (0, u). The last condition follows
from the fact that ∂t u, if it exists, is a mild solution of the differentiated
equation (93):
36
Jacek Banasiak
◦
∫t
◦
AGA (t − s)ϕ(s)ds
∂t u(t) = GA (t)A(u +f (0, u)) +
(94)
0
where ϕ(s) = f (s, u(s)) is Hölder continuous by the regularization property
mentioned above. Then continuity of ∂t u follows from Theorem 8.
3 Transport on graphs
Let us recall that we consider a system of equations

∂ u (x, t)
= cj ∂x uj (x, t), x ∈ (0, 1),


 t j
uj (x, 0)
= fj (x),
m
∑

−

ϕ+
 ϕij ξj cj uj (1, t) = wij
ik (γk ck uk (0, t)),
t ≥ 0,
(95)
k=1
−
(ϕ−
ij )1≤i≤n,1≤j≤m
and Φ+ = (ϕ+
where Φ =
ij )1≤i≤n,1≤j≤m are, respectively,
the outgoing and incoming incidence matrices, while γj > 0 and ξj > 0 are the
absorption/amplification coefficients at, respectively, the head and the tail of
the edge ej .
Theorem 21. The following conditions are equivalent:
1. (A, D(A)) generates a C0 semigroup;
2. Each vertex of G has an outgoing edge.
Proof. 1. ⇒ 2. Assume that there is a semigroup (TA (t))t≥0 generated by A
and consider a classical solution u(t) = TA (t)f with f ∈ D(A). Suppose that
a vertex, say, vi has no outgoing edge. Then, by (11),
0=
m
∑
ϕ+
ik γk ck uk (0, t),
t > 0.
k=1
In particular, uk (x, t) = fk (x + ck t) for 0 ≤ x + ck t ≤ 1 so uk (0, t) = f (ck t)
for 0 ≤ t ≤ c1k . Thus
0=
m
∑
ϕ+
ik γk ck fk (ck t),
0 ≤ t ≤ c−1 := min{c−1
k }.
k=1
There is a sequence (f r )r∈N , f r ∈ D(A), approximating 1 = (1, 1, . . . , 1) in X.
Then
−1
0 ≤ ∥1|(0,c−1 ) −
(fkr (ck ·))1≤k≤m ∥X
c∫
m
∑
1
=
ck
k=1
≤
m
∑
k=1
1
ck
∫1
|1 − fkr (z)|dz → 0
0
0
ck
|1 − fkr (z)|dz
Kinetic models in natural sciences
37
as r → ∞. Since X−convergence implies convergence almost everywhere of a
subsequence, we have
m
∑
0=
ϕ+
ik γk ck
k=1
−1
almost everywhere on (0, c ), and thus everywhere. Since the graph is connected and we assumed that there is no outgoing edge at vi , there must be an
incoming edge and thus at least one term of the sum is positive while all other
terms are nonnegative. Thus, if there is a vertex with no outgoing edge, then
the set of initial conditions satisfying the boundary conditions is not dense in
X and thus (A, D(A)) cannot generate a C0 -semigroup.
1. ⇒ 2. It can be proved that, under assumption 2., the boundary conditions can be incorporated into the domain of the operator in the following
compact form:
D(A) = {u ∈ (W11 ([0, 1]))m ; u(1) = K−1 C−1 BGCu(0)},
(96)
where B is the adjacency matrix defined in (9). Clearly, (C0∞ (]0, 1[))m ⊂ D(A)
and hence D(A) is dense in X. Let us consider the resolvent equation for A.
We have to solve
λuj − cj ∂x uj = fj ,
j = 1, . . . , m,
x ∈ (0, 1),
with u ∈ D(A). Integrating, we find the general solution
cj uj (x) = cj e
λ
cj
x
∫1
λ
e cj
vj +
(x−s)
fj (s)ds,
(97)
x
( λ )
s
where v = (v1 , . . . , vm ) is an arbitrary vector. Let Eλ (s) = diag e cj
.
1≤j≤m
Then (97) takes the form
∫1
Cu(x) = CEλ (x)v +
Eλ (x − s)f (s)ds.
x
To determine v such that u ∈ D(A), we use the boundary conditions. At
x = 1 we obtain
Cu(1) = CEλ (1)v
and at x = 0
∫1
Cu(0) = Cv +
Eλ (−s)f (s)ds
0
so that
38
Jacek Banasiak

KCEλ (1)v = KCu(1) = BGCu(0) = BG Cv +
∫1

Eλ (−s)f (s)ds ,
0
which can be written as
(I − Eλ (−1)C
−1
−1
K
BGC)v = Eλ (−1)C
−1
−1
K
∫1
BG
Eλ (−s)f (s)ds.
0
Since the norm of Eλ (−1) can be made as small as one wishes by taking
large λ, we see that v is uniquely defined by the Neumann series provided λ
is sufficiently large and hence the resolvent of A exists. We need to find an
estimate for it. First we observe that the Neumann series expansion ensures
that A is a resolvent positive operator and hence the norm estimates can
be obtained using only nonnegative entries. Next, we recall that B is column
stochastic; that is, each column sums to 1. Adding together the rows in
∫1
KCEλ (1)v = BGCv + BG
Eλ (−s)f (s)ds.
0
we obtain
m
∑
λ
cj
ξj cj e vj =
j=1
m
∑
γj cj vj +
j=1
m
∑
∫1
γj
j=1
− cλ s
e
j
fj (s)ds.
0
Integrating (97) for j ∈ {1, . . . , m}, we obtain
∫1
∫1
uj (x)dx = vj
0
e
λ
cj
x
1
dx +
cj
∫1 ∫1
0
0
λ
e cj
(x−s)
fj (s)ds
x
) 1 ∫1 (
)
vj cj ( cλj
−λs
=
e −1 +
1 − e cj fj (s)ds
λ
λ
0
so that, introducing a weighted space XΞ with the norm
∥u∥Ξ =
m
∑
j=1
we have
ξj ∥uj ∥L1 ([0,1]) ,
Kinetic models in natural sciences
∥u∥Ξ =
m
∑
39
∫1
ξj
j=1
uj (x)dx
(98)
0
∫1 (
m
m
( λ
) 1∑
)
1∑
−λs
cj
ξj vj cj e − 1 +
ξj
1 − e cj fj (s)ds
=
λ j=1
λ j=1
0
1
=
λ
m
∑
1
cj vj (γj − ξj ) +
λ
j=1
m
∑
j=1
∫1
(γj − ξj )
− cλ s
e
0
j
1∑
fj (s)ds +
ξj
λ j=1
m
∫1
fj (s)ds.
0
We consider three cases (the first one essentially coincides with [18, Proposition 3.3]).
(a) γj ≤ ξj for j = 1, . . . , m. Let us consider the iterates in the Neumann series
for v, (Eλ (−1)C−1 K−1 BGC)n . Using the fact that C, G and K are diagonal
so that they commute, we find
Eλ (−1)C−1 K−1 BGC ≤ (CK)−1 Eλ (−1)B(CK).
Since B is (column) stochastic, we have r(Eλ (−1)C−1 K−1 BGC) < 1 for any
λ > 0. Hence R(λ, A) is defined and positive for any λ > 0. Under the assumption of this item, by dropping two first terms in the second line, (98)
gives
∫1
m
1∑
1
∥u∥Ξ ≤
ξj fj (s)ds = ∥f ∥Ξ , λ > 0.
λ j=1
λ
0
Since D(A) is dense in X, (A, D(A)) generates a positive semigroup of contractions in X.
(b) γj ≥ ξj for j = 1, . . . , m. Then (98) implies that for some λ > 0 and
c = 1/λ we have
∥R(λ, A)f ∥Ξ ≥ c∥f ∥Ξ
and, by density of D(A), the application of the Arendt-Batty-Robinson theorem, Theorem 10, gives the existence of a positive semigroup generated by
A in XΞ . Since, however, the norm ∥ · ∥Ξ and the standard norm ∥ · ∥ are
equivalent, we see that A generates a positive semigroup in X.
(c) γj < ξj for j ∈ I1 and γj ≥ ξj for j ∈ I2 , where I1 ∩ I2 = ∅ and
I1 ∪ I2 = {1, . . . , m}. Let L = diag(lj ) where lj = ξj for j ∈ I1 and lj = γj for
j ∈ I2 . Then
Eλ (−1)C−1 K−1 BGC ≤ (CK)−1 Eλ (−1)B(CL).
Thus, if we denote by AL the operator given by the expression A restricted to
D(A) = {u ∈ (W11 ([0, 1]))m ; u(1) = K−1 C−1 BLCu(0)},
we see that
40
Jacek Banasiak
0 ≤ R(λ, A) ≤ R(λ, AL )
(99)
for any λ for which R(λ, AL ) exists. But, by item (b), AL generates a positive
semigroup and thus satisfies the Hille–Yosida estimates. Since clearly (99)
yields Rk (λ, A) ≤ Rk (λ, AL ) for any k ∈ N, for some ω > 0 and M ≥ 1 we
have
∥Rk (λ, A)∥ ≤ ∥Rk (λ, AL )∥ ≤ M (λ − ω)−k ,
λ>ω
and hence we obtain the generation of a semigroup by A.
Thus the proof of the theorem is complete.
⊓
⊔
4 Epidemiology
In this chapter, we simplify the SIRS model given by (17) and consider the
SIS model describing the evolution of epidemics which does not convey any
immunity. Setting γ = 0 in the system (17) and thus discarding the ‘recovered’
class, we have
∂t s(a, t) + ∂a s(a, t) = −µ(a)s(a, t) − Λ(a, i(·, t))s(a, t) + δ(a)i(a, t),
∂t i(a, t) + ∂a i(a, t) = −µ(a)i(a, t) + Λ(a, i(·, t))s(a, t) − δ(a)i(a, t),
∫ ω
s(0, t) =
β(a) {s(a, t) + (1 − q)i(a, t)} da,
0
∫ ω
i(0, t) = q
β(a)i(a, t) da,
(100a)
(100b)
(100c)
(100d)
0
s(a, 0) = s0 (a) = ϕs (a),
(100e)
i
i(a, 0) = i0 (a) = ϕ (a),
(100f)
for 0 ≤ t ≤ T ≤ +∞, 0 ≤ a ≤ ω ≤ +∞.
The force of infection is defined by (18) or (19); the concrete assumption
will be introduced when needed. In both cases we deal with a semilinear
problem; that is, with a nonlinear (algebraic) perturbation of a linear problem.
As in Section 2.7, the decisive role is played by the semigroup generated by
the linear part of the problem.
Problems like (100a)-(100f) have been relatively well-researched, including the cases where µ and β are nonlinear functions depending on the total
population, see [17, 31] and reference therein. Our model most resembles that
discussed in [31], the main difference being that in op. cit. the maximum age
ω is infinite, thus µ is bounded. However, a biologically realistic assumption
is that ω < +∞. This, however, necessitates building into the model a mechanism ensuring that no individual can live beyond ω. It follows, e.g. [21], that
the probability of survival of an individual till age a is given by
Π(a) = e−
Thus Π(ω) = 0 which requires
∫a
0
µ(s) ds
.
Kinetic models in natural sciences
∫
41
ω
µ(s) ds = +∞.
(101)
0
Hence, µ cannot be bounded as a → ω − . This is in contrast with the case
ω = +∞, where commonly it is assumed that µ is a bounded function on R+ ,
and introduces another unbounded operator in the problem. We note that
this difficulty was circumvented by Inaba in [22] by introducing the maximum
reproduction age a† < ω and ignoring the evolution of the post-reproductive
part of the population. Also, in papers such as [17], though ω < +∞, the
assumption that the population is constant removes the death coefficient from
the equation. The analysis of the model without any simplification in the scalar
and linear case was done in [21] by reducing it to an integral equation along
characteristics. It can be proved that the solution of such a problem is given by
a strongly continuous semigroup. Here we shall prove this directly by refining
the argument of [22].
Notation and assumption
We will work in the space X = (L1 ([0, ω]))2 with norm ∥(p1 , p2 )∥X = ∥p1 ∥ +
∥p2 ∥, where the norm ∥·∥ refers to the norm in L1 ([0, ω]); the relevant norm
in R2 will be denoted by | · |. We also introduce necessary assumptions (cf.
[21]) on the coefficients of (100).
(
)
(H1) 0 ≤ µ ∈ L∞,loc [0, ω) , satisfying (101), with 0 < µ ≤ µ(a);
(
)
(H2) 0 ≤ β ∈ L∞ [0, ω] with β(a) ≤ β := esssup β(a);
a∈[0,ω]
(
)
(H3) 0 ≤ δ ∈ L∞ ([0, ω] with
δ
≤
δ(a)
≤
δ;
)
(H4) 0 ≤ K ∈ L∞ [0, ω]2 .
(
)
functions.
FurLet (W11 [0, ω] )2 be the Sobolev space of vector valued
(
)
ther, we define S = diag {−∂a , −∂a } on D(S) = (W11 [0, ω] )2 , Mµ (a) =
diag {−µ(a), −µ(a)} on D(Mµ ) = {φ ∈ X : µφ ∈ X},
(
)
0 δ(a)
Mδ (a) =
;
(102)
0 −δ(a)
Mδ ∈ L(X). Further
(
B(a) =
β(a) (1 − q)β(a)
0
qβ(a)
∫
with
)
(103)
ω
Bφ =
B(a)φ(a) da;
0
the operator B is bounded. Moreover, we introduce the linear operator Aµ
defined on the domain
42
Jacek Banasiak
D (Aµ ) = {φ ∈ D(S) ∩ D(Mµ ); φ(0) = Bφ}
(104)
Aµ = S + Mµ .
(105)
by
Let Q be the linear operator defined on the domain D(Q) = D(Aµ ) by
Q = Aµ + Mδ .
Using these notations, we re-write (100a)-(100f) in the following compact form
∂t u = Aµ u + Mδ u + F(u),
∫ ω
B(a)u(a, t) da,
u(0, t) =
(106a)
(106b)
0
u(a, 0) = u0 (a) = φ(a),
(106c)
where u = (s, i)T and F is a nonlinear function defined by
(
)
( )
−Λ(u) 0
F u =
,
Λ(u) 0
4.1 The linear part
To prove that (100a)-(100f) is well-posed in X, first we show that the linear operator Q on D(Q) = D(Aµ ) generates a strongly continuous positive
semigroup on X.
Theorem 4.1. The linear operator Q generates a strongly continuous positive
semigroup (TQ (t))t≥0 in X.
To carry out the proof of Theorem 4.1, it is sufficient to prove the generation result for Aµ and use Theorem 12 (the bounded perturbation theorem)
to prove the generation for Q; then we use some other tools to show the
positivity of the combined semigroup
In this setting, the following result holds.
Lemma 4.2.
operator Aµ generates a strongly continuous positive
( The linear
)
semigroup TAµ (t) t≥0 in X such that
∥T Aµ (t)∥L(X) ≤ e(β−µ)t .
(107)
To prove the lemma we construct and estimate the resolvent of Aµ . First
we introduce the survival rate matrix L(a), which represents the survival
rate function in a multi-state population. L(a) is a solution of the matrix
differential equation:
dL
(a) = Mµ (a)L(a),
da
L(0) = I,
(108)
Kinetic models in natural sciences
43
where I denotes the 2 × 2 identity matrix. The solution of (108) is a diagonal
matrix given by
∫a
L(a) = e− 0 µ(r) dr I.
(109)
We see that
∫a
L−1 (a) = e
0
µ(r) dr
I.
(110)
Hence, we can define the resolvent (or state transition matrix, or principal
fundamental matrix) L(a, b) as
L(a, b) = L(a)L−1 (b).
−1
Lemma 4.3. If λ > β − µ, then (λI − Aµ )
is given by
−1
φ = (λI − Aµ ) ψ
(
)−1 ∫ ω
∫ ω
∫ a
e−λa B(a)L(a)
= e−λa L(a) I −
e−λσ B(σ)L(σ) dσ
eλσ L−1 (σ)ψ(σ) dσ da
0
0
0
∫ a
+ e−λa L(a)
eλσ L−1 (σ)ψ(σ) dσ.
(111)
0
Proof. Let λ > β − µ and ψ ∈ X. A function φ ∈ D(Aµ ) if and only if
λφ(a) +
d
φ(a) − Mµ (a)φ(a) = ψ(a),
da
∫
(112a)
ω
φ(0) =
B(a)φ(a) da,
(112b)
0
µφ ∈ X.
By Duhamel’s formula, (112a) leads to
∫ a
φ(a) = e−λa L(a)φ(0) +
e−λ(a−s) L(a, s)ψ(s) ds,
0
∫ a
∫
∫a
−λa− 0a µ(r) dr
=e
φ(0) +
e−λ(a−s)− s µ(r) dr ψ(s) ds,
(112c)
(113)
0
for some unspecified as yet initial condition φ(0). For a fixed φ(0) we denote
φ := Rφ(0) (λ)ψ; it is easy to see that
(λI − S − Mµ )Rφ(0) (λ)ψ = ψ,
(114)
for a.a. a ∈ [0, ω). The unknown φ(0) can be determined from (112b) by
substituting (113); we get
∫ ω
∫a
φ(0) =
e−λa− 0 µ(r) dr B(a)φ(0) da
0
(∫ a
)
∫ ω
∫
∫
−λa− 0a µ(r) dr
λs+ 0s µ(r) dr
+
e
B(a)
e
ψ(s) ds da.
0
0
44
Jacek Banasiak
Since
∫
ω
−λa−
e
∫a
0
µ(r) dr
0
∫
B(a) da ≤ β
ω
e−(λ+µ)a da ≤
0
β
<1
λ+µ
(115)
∫a
∫ω
for λ > β̄ − µ, I − 0 e−λa− 0 µ(r) dr B(a) da is invertible with
(
)−1 ∫ ω
∫
λ+µ
−λs− 0s µ(r) dr
.
e
B(s) ds
I−
≤
λ − (β − µ)
0
(116)
Hence
(
)−1
∫ ω
∫
−λa− 0a µ(r) dr
e
φ(0) = I −
B(a) da
0
(∫
)
∫ ω
a
∫
∫
−λa− 0a µ(r) dr
λs+ 0s µ(r) dr
e
B(a)
e
ψ(s) ds da
0
0
and we can substitute that φ(0) in the operator Rφ(0) (λ) to define
R(λ)ψ(a)
−λa−
:= e
∫a
0
µ(r) dr
(
∫
I−
ω
∫
−λs− 0s µ(r) dr
e
)−1 ∫
0
∫
a
×
eλs+
∫s
0
µ(r) dr
∫a
ψ(s) ds da + e−λa−
0
ω
B(s) ds
0
e−λa−
∫a
0
µ(r) dr
B(a)
0
∫
a
µ(r) dr
eλs+
∫s
0
µ(r) dr
ψ(s) ds.
0
The above calculations show that λ − Aµ is one-to-one for λ > β − µ. Routine
calculations show that
R(λ)ψ ≤
X
1
ψ .
X
λ − (β − µ)
Thus R(λ) is a bounded operator in X.
Further, we also show that
(
∫
ω
|µ(a)R(λ)ψ(a)| da ≤
0
β
1+
λ − (β − µ)
(117)
)
ψ .
X
Hence R(λ)X ⊂ D(Mµ ). Further, since for any ψ ∈ X, φ = R(λ)ψ satisfies
λR(λ)ψ +
d
R(λ)ψ − Mµ R(λ)ψ = ψ
da
almost everywhere, we have
d
R(λ)ψ = ψ − λR(λ)ψ + Mµ R(λ)ψ,
da
Kinetic models in natural sciences
45
where, by the above (estimates, all terms on the right hand side are in X.
Hence R(λ)ψ ∈ (W11 [0, ω]))2 .
Since the boundary condition holds, using the results above, we see that
R(λ) : X → D(Aµ ). Then R(λ) is a right-inverse of the operator λ−A, hence
R(λ) is the right inverse of (λI − Aµ , D(Aµ )). To prove that it is also a left
inverse, we repeat the standard argument. Assume that for some φ ∈ D(Aµ )
we have
e ̸= φ.
R(λ)(λI − Aµ )φ = φ
Since R(λ) : X → D(Aµ ), we can write
(λI − Aµ )φ = (λI − Aµ )R(λ)(λ − Aµ )φ = (λ − Aµ )e
φ
since R(λ) is a right inverse of λI −Aµ . But we proved that the linear operator
e and hence R(λ) = (λ − Aµ )−1 .
(λ − Aµ ) is one-to-one for λ > β − µ, φ = φ
Lemma 4.4. D(A)+ = X+ .
Proof. A proof of this result (with gaps) is provided in [22, p. 60]. A more
comprehensive proof can be found in [31]. We present a much simpler proof
which, moreover, allows for an approximation of ψ ∈ X+ by elements of
D(Aµ )+ .
Fix f ∈ X+ . First we note that for any given ϵ there is 0 ≤ ϕ ∈
(C0∞ (]0, ω[))2 such that ∥f − ϕ∥X ≤ ϵ. Clearly, ϕ ∈ D(Mµ ) but typically
∫ ω
φ(0) ̸=
B(a)φ(a) da.
0
C0∞ ([0, ω))
with η(0) = 1 and let ηϵ (a) = η(a/ϵ).
Take a function 0 ≤ η ∈
Further, let α = (α1 , α2 ) be a vector and consider
ψ = φ + ηϵ α.
Clearly, ψ ∈ (W11 ([0, ω]))2 ∩ D(Mµ ). As far as the boundary condition is
concerned we have, by the properties of the involved functions,
 ϵω

)
∫ ω
∫ (
β(a)ηϵ (a) (1 − q)β(a)ηϵ (a)
α=
B(a)φ(a) da + 
da α. (118)
0
qβ(a)ηϵ (a)
0
0
Now, since
∫ϵω
0≤
∫ω
the matrix l1 −norm satisfies
 ϵω
∫ (
β(a)ηϵ (a)

0
0
β(ϵs)η(s)ds ≤ ϵβ̄,
β(a)ηϵ (a)da = ϵ
0
0
(1 − q)β(a)ηϵ (a)
qβ(a)ηϵ (a)
)


da ≤ ϵβ̄.
L(R2 )
46
Jacek Banasiak
Thus, (118) is solvable for sufficiently small ϵ, by positivity of the above matrix
and the Neumann series representation, α is nonnegative and
∫ ω
|α| ≤ B(a)φ(a) da (1 − ϵβ̄)−1 ≤ C
0
for some constant C, which is independent of ϵ for sufficiently small ϵ (∥φ∥,
which depends on ϵ, can be bounded by e.g. ∥f ∥ + 1 for ϵ < 1). Hence we have
∥f − ψ∥X ≤ ∥f − φ∥X + |α|∥ηϵ ∥ ≤ (1 + C)ϵ.
Proof of Lemma 4.2. Since the inverse of a bounded operator is closed, we
see that λI − Aµ , and hence Aµ , are closed. Thus the above lemmas with
the estimate (117) show that Aµ satisfies the assumptions of the Hille–Yosida
theory. Hence, it generates a semigroup satisfying (107). Since the resolvent
is positive, the semigroup is positive as well. ⊓
⊔
Proof of Theorem 4.1. Since Mδ ∈ L (X), with Mδ (a)L(X) ≤ 2δ, Theorem
12 (the bounded perturbation theorem) is applicable and states that the linear
(operator) (Q, D(A)) generates a strongly continuous semigroup denoted by
T Q (t) t≥0 . Using the estimate (107) we have:
∥T Q (t)∥L(X) ≤ et(β−µ+2δ̄) .
Using the structure of Mδ we can improve this estimate and also show that
the semigroup (T Q (t))t≥0 is positive. Since the the variable a plays in Mδ
the role of a parameter, we find
(
)
1 1 − e−tδ(a)
T Mδ (t) =
,
0
e−tδ(a)
and so we have
∥T Mδ (t)∥L(X) = 1.
Also, (T Mδ (t))t≥0 is positive. Hence, by (76), we obtain
∥T Q (t)∥L(X) ≤ et(β−µ)
(119)
⊓
⊔
and (T Q (t))t≥0 is positive.
Remark 6. The estimates (107) and (119) are not optimal. In fact, for the
scalar problem
∂t n(a, t) = −∂a n(a, t) − µ(a)n(a, t),
∫ω
n(0, t) = β(a)n(a, t)da,
0
◦
n(a, 0) = n (a),
t > 0, a ∈]0, ω[
(120)
Kinetic models in natural sciences
47
it can be proved, [21], that there is a unique dominant eigenvalue λ∗ of the
problem such that
∗
∥n(t)∥ ≤ M etλ ,
(121)
which is the solution of the renewal equation
∫ω
1=
∫a
−λa− µ(s)ds
β(a)e
0
da.
(122)
0
This eigenvalue is, respectively, positive, zero or negative if and only if the
basic reproduction number
∫ω
R=
−
β(a)e
∫a
0
µ(s)ds
da
(123)
0
is bigger, equal, or smaller, than 1.
◦ ◦
Consider now an initial condition (s, i ) ∈ D(Aµ )+ . Since the semigroup
(T Q (t))t≥0 is positive, the strict solution (s, i) of the linear part of (100) is
nonnegative and the total population 0 ≤ s(a, t) + i(a, t) = n(a, t) satisfies
(120). Using nonnegativity, we find s(a, t) ≤ n(a, t) and i(a, t) ≤ n(a, t) and
consequently
∗
◦ ◦
◦ ◦
∥T Q (t)( s, i )∥X ≤ M etλ ∥(s, i )∥X
◦ ◦
for (s, i ) ∈ D(Aµ )+ . However, by Lemma 4.4, the above estimate can be
extended to X+ and, by (29), to X.
Note that the crucial role in the above argument is played by the fact that
(s, i) satisfies the differential equation (100) — if it was only a mild solution,
it would be difficult to directly prove that the sum s + i is the mild solution
to (120).
4.2 The nonlinear problem
Intercohort Transmission
In the case of intercohort transmission, individuals of any age can infect individuals of any age, though with possibly different intensity. Then
∫ ω
Λ(a, i) =
K(a, a′ )i(a′ ) da′ ,
(124)
0
′
where K(a, a ) is a nonnegative bounded function on [0, ω] × [0, ω] which
account for the age-specific (average) probability of becoming infected through
contact with infectives of a particular age.
Since the nonlinear term F is quadratic, it is easy to see that the following
result is true.
48
Jacek Banasiak
Proposition 4.5. F is( continuously
)
(Fréchet) differentiable with respect to
ϕ ∈ X and for any ϕ = ϕs , ϕi , ψ = ψ s , ψ i ∈ X the Fréchet derivative at
ϕ, Fϕ , is defined by


∫ω
∫ω
s
′
i
′
′
s
′
i
′
′
−ψ
(a)
K
(a,
a
)
ϕ
(a
)
da
−
ϕ
(a)
K
(a,
a
)
ψ
(a
)
da


(
)
0
0



.
Fϕ ψ (a) := 

ω
∫ω
∫
 s

′
i
′
′
s
′
i
′
′
ψ (a) K (a, a ) ϕ (a ) da + ϕ (a) K (a, a ) ψ (a ) da
0
0
(125)
◦
◦ ◦
This allows for applying Theorem 19 to claim that for each u = (s, i ) ∈ X,
◦
there is a t(u) such that the problem (106) has a unique mild solution on
◦
◦
[0, t(u)[∋ t → u(t); this solution is a classical solution if u ∈ D(Aµ ).
We recall that that the proof consists of showing that the Picard iterates
◦
u0 = u
◦
∫t
un (t) = T Q (t) u +
T Q (t − s)F(un−1 (s))ds
(126)
0
◦
◦
◦
◦
converge in C([0, t(u)[, B(u, ρ)) where B(u, ρ) = {u ∈ X; ∥u− u ∥X ≤ ρ} for
some constant ρ. Since the nonlinearity is quadratic, it is not globally Lipschitz
continuous and thus the question whether this solution can be extended to
[0, ∞[ requires employing positivity techniques.
Since F is not positive on X+ , we cannot claim that the constructed local
solution is nonnegative, as the iterates defined by (126) need not to be positive,
◦
even if we start with u≥ 0. Hence, we re-write (106) in the following equivalent
form

( )
 du
= (Q − κI) u + (κI + F) u , t > 0,
(127)
dt
◦
 u(0) = u,
for some κ ∈ R+ to be determined. Denote Qκ = Q−κI; then (T Q,κ (t))t≥0 =
(e−κt T Q )t≥0 and hence (T Q,κ (t))t≥0 is positive.
Thus the mild solution satisfies
∫ t
(
)
◦
◦
u(t) = e−κt T (t) u +
e−κ(t−s) TQ (t − s) κI + F (u(s)) ds, 0 ≤ t < t(u).
0
(128)
It is easy to see that the following result holds.
(
)
◦
Lemma 4.6. For any ρ there exists κ such that κI+F (X+ ∩B(u, ρ)) ⊂ X+ .
With this result, the Picard iterates for (128) are nonnegative and we can
repeat the standard estimates to arrive at the following result:
Kinetic models in natural sciences
◦
49
◦
Theorem 4.7. Assume that u∈ X+ and let u : [0, t(u)[→ X be the unique
mild solution of (106). Then this solution is nonnegative on the maximal interval of its existence. Moreover, the solutions continuously depend on the initial
conditions on every compact time interval of their joint interval of existence.
4.3 Global existence
Since quadratic nonlinearities do not satisfy the uniform Lipschitz condition,
we cannot immediately claim that the solutions to (106) are global in time.
In fact, it is well known that, even for ordinary differential equations, the
solution with a quadratic nonlinearity can blow in finite time. Here, we use
positivity to show thatpositive
solutions exist globally in time. For this, we
have to show that t → u(t)X does not blow up in finite time. We state the
following result:
◦
Theorem 4.8. For any u∈ D(Aµ ) ∩ X+ , the problem (106) has a unique
strict positive solution u(t) defined on the whole time interval [0, ∞[.
Proof. The proof is follows the same idea as in Remark 6. Under( the assump)
tions, we have locally defined positive strict solution u(t) = s(t), i(t) to
(106) and hence of (100) in (L1 ([0, ω]))2 . Thus
∫ ω
∫ ω
u(t) =
(s(a, t) + i(a, t)) da =
u(a, t) da,
X
0
0
where u(a, t) is the solution to the McKendrick equation (120). But then, as
◦
long as for 0 ≤ t < t(u),
◦
u(t) ≤ e(β−µ)t u
.
X
X
Accordingly, u(t)X does not blow up in finite time and hence the solution
is global.
◦
Corollary 2. For any u∈ X+ , the problem (106) has a unique mild positive
solution u(t) defined on the whole time interval [0, ∞[.
The proof follows from D(Aµ )+ = X+ and the continuous dependence on
initial conditions.
4.4 Intracohort Transmission
In this section, we are concerned with the situation where the disease
transmission interactions are restricted to individuals of the same range of
age. A constitutive form of the infection rate relative to this mechanism of
transmission is provided in [16, p. 1381]:
Λ(a, i(·, t)) = K0 (a)i(a, t).
(129)
50
Jacek Banasiak
Therefore, the nonlinear term (also known as forcing term) F in (106) is
defined by
[
]
(
)
−K0 (a)s(a, t)i(a, t)
F u(a, t) =
, u = (s, i).
K0 (a)s(a, t)i(a, t)
In the sequel, we use the notation below.
X1 = L1 ([0, ω], R2 ),
X∞ = L∞ ([0, ω], R2 ),
(
)
Y∞ = C [0, T ], X∞ ,
The main problem with the intracohort transmission is that, in general,
F(u) ∈
/ X1 for u ∈ X1 . Multiplication is well defined in L∞ ([0, ω]) but then
the latter space is not suitable for semigroup techniques — any C0 -semigroup
on L∞ is uniformly continuous. To handle this nonlinearity, we use the fact
that for ω < ∞, X∞ is a subspace densely and continuously embedded in X1
and show that we can restrict the analysis performed in the previous section
to X∞ .
First we note the following result.
◦
Proposition 4.9. For any t ≥ 0, TQ (t) (X∞ ) ⊂ X∞ with ∥TQ (t) u∥X∞ ≤
◦
◦
βωeβω ∥u∥X∞ for all u∈ X∞ .
This result follows from the integral
representation of the semigroup.
)
Now, let u(t) = (s(·, t), i(·, t) be a function in X∞ . Under the above
setting, (100a)-(100f) is rewritten, as in the previous section, in the following
abstract differential equation in a Banach space X∞ :

 du
= Qu + F (u) ,
t > 0,
(130)
dt
◦
 u(0) =u
.
We observe that the nonlinear function F has the following properties:
( )
• F(X∞ ) ⊂ X∞ with ∥F u ∥X∞ ≤ ∥K0 ∥∞ ∥u∥2X∞ ;
• F is continuously Fréchet differentiable with respect to φ ∈ X∞ hence, in
particular, it is locally Lipschitz continuous.
Using this properties and Proposition 4.9 we can show that Picard iterates of
the integral formulation of (130):
∫ t
(
)
◦
(u) (t) = TQ (t) u +
TQ (t − s)F u(s) ds, 0 < t < T,
(131)
0
stay in an an appropriately chosen closed ball in Y∞ for sufficiently small
T > 0 and converge to a unique fixed point which is thus a mild solution of
(130).
Kinetic models in natural sciences
51
Restricting considerations to such a ball, we can consider the problem with
the linear L1 semigroup to justify calculations and, in the same way as in the
intercohort case, prove positivity, global existence and classical solvability for
◦
u∈ D(Aµ ) ∩ X∞ .
5 Coagulation-fragmentation equation
Recall that we deal with the equation
∫∞
∂t u(x, t) = −a(x)u(x, t) +
a(y)b(x|y)u(y, t)dy
x
∫∞
−u(x, t)
k(x, y)u(y, t)dy
0
+
1
2
∫x
k(x − y, y)u(x − y, t)u(y, t)dy,
(132)
0
where x ∈ R+ :=]0, ∞[ is the size of the particles/clusters. Here u is the
density of particles of mass/size x, a is the fragmentation rate and b describes
the distribution of masses x of particles spawned by fragmentation of a particle
of mass y.
Thanks to the work of P. Laurençot, [26], there is a well developed theory
of weak solvability of coagulation–fragmentation problems. The main aim of
this section is to present results on strict solvability of them.
Fragmentation rates.
The fragmentation rate a is assumed to satisfy
0 ≤ a ∈ L∞,loc ([0, ∞[),
(133)
that is, we allow a to be unbounded as x → ∞. Further, b ≥ 0 is a measurable
function satisfying b(x|y) = 0 for x > y and (24).
The expected number of particles resulting from a fragmentation of a size
y parent, denoted by n0 (y), is assumed to satisfy
n0 (y) < +∞
(134)
for any fixed y ∈ R+ .
We impose some control on the growth of the fragmentation rates. Namely,
we assume that there are j ∈]0, ∞[, l ∈ [0, ∞[ and a0 , b0 ∈ R+ such that for
any x ∈ R+
a(x) ≤ a0 (1 + xj ),
n0 (x) ≤ b0 (1 + xl ).
(135)
52
Jacek Banasiak
Coagulation rates.
The coagulation kernel k(x, y) represents the likelihood of a particle of size
x attaching itself to a particle of size y. We assume that it is a measurable
symmetric function such that for some K > 0 and 0 ≤ α < 1
0 ≤ k(x, y) ≤ K((1 + a(x))α (1 + a(y))α .
(136)
This will suffice to show local in time solvability of (132) whereas to show
that the solutions are global in time we need to strengthen (136) to
0 ≤ k(x, y) ≤ K((1 + a(x))α + (1 + a(y))α )
(137)
for some 0 ≤ α < 1.
State spaces.
In fragmentation and coagulation problems, two spaces are most often used
due to their physical relevance.∫ In the space L1 (R+ , xdx) the norm of a non∞
negative element u, given by 0 u(x)xdx, represents the total mass of the
system,
whereas the norm of a nonnegative element u in the space L1 (R+ , dx),
∫∞
u(x)dx, gives the total number of particles in the system.
0
We use the scale of spaces with finite higher moments
Xm = L1 (R+ , dx) ∩ L1 (R+ , xm dx) = L1 (R+ , (1 + xm )dx),
(138)
where m ∈ M := [1, ∞[. We extend this definition to X0 = L1 (R+ ). We note
that, due to the continuous injection Xm ,→ X1 , m ≥ 1, any solution in Xm
is also a solution in the basic space X1 .
Thus, we denote by ∥ · ∥m the natural norm in Xm defined in (138). To
shorten notation, we define
wm (x) := 1 + xm .
5.1 Main Results
To formulate these results, we have to introduce specific assumptions and
notation. First we define
∫y
b(x|y)xm dx
nm (y) :=
0
for any m ∈ M0 := {0} ∪ M and y ∈ R+ . Further, let
N0 (y) := n0 (y) − 1
and
Nm (y) := y m − nm (y),
m ≥ 1.
Kinetic models in natural sciences
53
It follows that
N0 (y) = n0 (y) − 1 ≥ 0
and
∫y
Nm (y) = y
m
∫y
−
b(x|y)x dx ≥ y
m
m
−y
m−1
0
b(x|y)xdx = 0
(139)
0
for m ≥ 1 with N1 = 0.
The fragmentation part
Next, for any m ∈ M, let (Am u)(x) := a(x)u(x) on
D(Am ) = {u ∈ Xm : au ∈ Xm }
and let Bm be the restriction to D(Am ) of the integral expression
∫∞
[Bu](x) =
a(y)b(x|y)u(y)dy.
x
Theorem 22. Let a, b satisfy (24), (134) and (135), and let m be such that
m ≥ j + l if j + l > 1 and m > 1 if j + l ≤ 1.
a) The closure (Fm , D(Fm )) = (−Am + Bm , D(Am )) generates a positive
quasi-contractive semigroup, say (SFm (t))t≥0 , of type at most 4a0 b0 on Xm .
Furthermore, if u ∈ D(Fm )+ , then
Nm (x)a(x)u(x) ∈ X0 ,
m ∈ M0 .
(140)
b) If, moreover, for some m there is cm > 0 such that
lim inf
x→∞
Nm (x)
= cm ,
xm
(141)
then Fm = −Am + Bm and (SFm (t))t≥0 is an analytic semigroup on Xm .
c) If (141) holds for some m0 , then it holds for all m ≥ m0 .
We note that (141) cannot hold for m = 1 as N1 = 0.
Proof. We shall fix m satisfying m ≥ j + l if j + l > 1 and m > 1 otherwise;
see (135).
First we show that Bm := B|D(Am ) is well defined.
Next, direct integration gives for u ∈ D(Am )
∫∞
(−Am + Bm )u(x)wm (x)dx = −ϕm (u)
0
∫∞
:= (N0 (x) − Nm (x))a(x)u(x)dx,
0
(142)
54
Jacek Banasiak
If the term N0 (x) > 0 had not been present, then (142) would have allowed
a direct application of the substochastic semigroup theory. In the present case
we note that u ∈ D(Am )+ we have, by (139),
∫∞
−ϕm (u) ≤
∫∞
N0 (y)a(y)u(y)dy ≤ 4a0 b0
0
u(x)wm (x)dx =: η∥u∥m ,
0
∫∞
Then we have ϕem (u) := ϕm (u) + η 0 u(x)wm (x)dx ≥ 0 for 0 ≤ u ∈ D(Am )
em , D(Am )) := (Am + ηI, D(Am )) satisfies
and the operator (A
∫∞
em + Bm )u(x)wm (x)dx = −ϕem (u)
(−A
0
∫
∞
= −η
0
∫∞
u(x)wm (x)dx + (N0 (x) − Nm (x))a(x)u(x)dx
0
≤ 0.
em +Bm generates a substochastic semigroup (S e (t))t≥0
An extension Fem of −A
Fm
and thus there is an extension Fm of (−Am +Bm , D(Am )) given by (Fm , D(Fm )) =
(Fem +ηI, D(Fem )) generating a positive semigroup (SFm (t))t≥0 = (eηt SFem (t))t≥0
on Xm .
Furthermore, ϕem extends to D(Fm ) by monotone limits of elements of
D(Am ). Thus, let u ∈ D(Fm )+ with D(Am ) ∋ un ↗ u. Then, since
∫∞
∫∞
N0 (x)a(x)u(x)dx < ∞,
u(x)wm (x)dx < ∞,
0
0
by (135), m ≥ j + l and D(Fm ) ⊂ Xm , and the fact that ϕem (un ) tends to a
finite limit, we have
∫∞
lim
∫∞
Nm (x)a(x)un (x)dx =
n→∞
0
Nm (x)a(x)u(x)dx < +∞.
0
To show that (Fm , D(Fm )) = (−Am + Bm , D(Am )) we use Theorem 17. In
can be proved, [6, Theorem 6.20], that the generator Fm satisfies Fm ⊂ Fm,max
where Fm,max = A + B restricted to
D(Fm,max ) = {u ∈ Xm ; Au + Bu ∈ Xm }.
Note that neither Au nor Bu are guaranteed to be in Xm . Since au ∈
L1 ([0, R], (1 + xm )dx) for any 0 < R < +∞, therefore the same is true for Bu.
Hence, with wm (x) = 1 + xm ,
Kinetic models in natural sciences
55
∫∞
(−a(x)u(x) + [Bu](x)) wm (x)dx
0

= lim −
∫R
∫R
a(x)u(x)wm (x)dx +
R→∞
0


∫∞


a(y)b(x|y)u(y)dy  wm (x)dx .
x
0
Next, by (24),




∫R ∫∞
∫R ∫y
 a(y)b(x|y)u(y)dy  wm (x)dx =  b(x|y)wm (x)dx u(y)a(y)dy
x
0
∫∞
+
R


∫R

0
b(x|y)wm (x)xdx u(y)a(y)dy =
0
0
∫R
a(y)u(y)(n0 (y) + nm (y))ydy + SR ,
0
where SR ≥ 0. Combining this with (142), we see that
∫∞
(−a(x)u(x) + [Bu](x)) wm (x)dx
0
∫∞
= (N0 (x) − Nm (x))a(x)u(x)dx + lim SR ≥ −ϕm (u),
R→∞
0
so that Theorem 17 gives the thesis.
To prove part b), we begin by observing that inequality (139) implies that
0 ≤ Nm (x) ≤ xm . This, together with (141), yields cm xm /2 ≤ Nm (x) ≤ xm
for large x which, by (140), establishes that if u ∈ D(Fm ), then au ∈ Xm or,
in other words, that D(Fm ) ⊂ D(Am ). Since (Fm , D(Fm )) is an extension of
(−Am + Bm , D(Am )), we see that D(Fm ) = D(Am ).
It is clear that the semigroup generated by −Am is bounded. Furthermore,
if λ = r + is, then |λ + a(x)|2 ≥ s2 and therefore, for all r > 0
∥R(r + is, −Am )f ∥m
∫∞ 1
|f (x)|(1 + xm )dx ≤ 1 ∥f ∥m .
= r + is + a(x) |s|
0
The analyticity of the fragmentation semigroup then follows from the ArendtRhandi theorem.
Example 7. One of the forms of b(x|y) most often used in applications is
( )
1
x
b(x|y) = h
(143)
y
y
56
Jacek Banasiak
which is referred to as the homogeneous fragmentation kernel. In this case
the distribution of the daughter particles does not depend directly on their
relative sizes but on their ratio. In this case
∫y ( )
∫1
1
x
m
m
nm (y) =
h
x dx = y
h(z)z m dz =: hm y m .
y
y
0
0
Since
1
y = n1 (y) =
y
∫y ( )
∫1
x
h
xdx = y h(z)zdz = h1 y
y
0
0
we have h1 = 1 so that hm < 1 for any m > 1 and Nm (y) = y m (1 − hm ).
Hence, (141) holds.
On the other hand, fragmentation processes in which daughter particles
tend to accumulate close both to 0 and to the parent’s size may not satisfy
(141).
The coagulation part
Next, we introduce a nonlinear operator Cm in Xm defined for u from a
suitable subset of Xm by the formula
∫∞
(Cm u)(x) := −u(x)
1
k(x, y)u(y)dy +
2
0
∫x
k(x − y, y)u(x − y)u(y)dy
0
so that the initial value problem for (132) can be written as an abstract
semilinear Cauchy problem in Xm
∂t u = −Am u + Bm u + Cm u,
◦
u(0) = u .
(144)
To formulate the main theorems we have to introduce a new class of spaces
which, as we shall see later, is related to intermediate spaces associated with
the fragmentation operator Fm and its fractional powers. We set
{
}
∫ ∞
(α)
α
m
Xm := u ∈ Xm ;
|u(x)|(ω + a(x)) (1 + x ) dx < ∞ ,
(145)
0
where ω is a sufficiently large constant. Then we have
Theorem 23. Assume that a, b, k satisfy (24), (134), (135), (136)and (141)
◦
(α)
for some m0 > 1, and let m ≥ max{j + l, m0 } hold. Then, for each u ∈ Xm,+ ,
there is τ > 0 such that the initial value problem (144) has a unique nonnega(α)
tive classical solution u ∈ C([0, τ ], Xm ) ∩ C 1 ((0, τ ), Xm ) ∩ C((0, τ ), D(Am )).
Furthermore, there is a measurable representation of u which is absolutely
continuous in t ∈ (0, τ ) for any x ∈ R+ and which satisfies (132) almost
everywhere on R+ × (0, τ ).
Kinetic models in natural sciences
57
Finally, for global in time solvability we need to restrict the growth rate
of k. Namely, we have
Theorem 24. Let the assumptions of Theorem 23 hold with β = 0, that is,
let k satisfy (137). Furthermore, let the constant j from assumption (135) be
such that αj ≤ 1. Then any local solution of Theorem 23 is global in time.
Interlude – intermediate spaces associated with Fm .
Define
Fm,ω := Fm − ωI, Am,ω := Am + ωI,
D(Fm,ω ) = D(Fm ) = D(Am ) = D(Am,ω ),
(146)
where ω > 4a0 b0 is a fixed constant. The operators (Fm,ω , D(Am )) and
(−Am,ω , D(Am )) generate analytic semigroups (SFm,ω (t))t≥0 = (e−ωt SFm (t))t≥0
and (S−Am,ω (t))t≥0 = (e−ωt S−Am (t))t≥0 on Xm . Since each operator is invertible, the norms ∥u∥m,A := ∥Am,ω u∥m and ∥u∥m,F := ∥Fm,ω u∥m , u ∈ D(Am )
are equivalent to each other and also to the corresponding graph norms on
D(Am ). Then we have (up to the equivalence of the respective norms)
DFm,ω (α, r) = D−Am,ω (α, r).
(147)
We find it most convenient to use D−Am,ω (α, 1) which equals the real interpolation space (Xm , D(Am,ω ))α,1 . It follows that
(α)
(Xm , D(Am,ω ))α,1 =: Xm
{
}
∫ ∞
α
m
= u ∈ Xm ; ∥u∥(α)
=
|u(x)|(ω
+
a(x))
(1
+
x
)
dx
<
∞
,
m
0
Indeed, Let u ∈ Xm . Then, by the Fubini-Tonelli theorem and routine changes
of variables, we obtain


ω+a(x)
∫ ∞
∫


∥t−α Am,ω S−Am,ω (t)u∥L1 (J) =
|u(x)|(ω+a(x))α 
s−α e−s ds wm (x)dx
0
0
with
∫ω
0<
−α −s
s
0
e
ω+a(x)
∫
ds ≤
−α −s
s
e
∫∞
ds ≤
0
s−α e−s ds = Γ (1 − α).
0
In other words, there is a constant c1 ≥ 1 such that
(α)
(α)
c−1
1 ∥u∥m ≤ ∥u∥DFm,ω (α,1) ≤ c1 ∥u∥m ,
∀ u ∈ DFm,ω (α, 1).
(148)
58
Jacek Banasiak
Proof of local solvability
Here we assume that a and b satisfy the assumptions of Theorem 22 b) so
that, in particular, (141) holds for some m ≥ j + l or m > 1 if j + l ≤ 1.
Furthermore, the coagulation kernel is such that (136) is satisfied. We fix
α
ω > max{4a0 b0 , 1} and denote aα
ω (x) := (ω + a(x)) .
We consider the following modified version of (132)
∫∞
∂t u(x, t) = −(aω (x) + γaα
ω (x))u(x, t) +
a(y)b(x|y)u(y, t)dy
x
∫∞
+(γaα
ω (x)
+ ω)u(x, t) − u(x, t)
k(x, y)u(y, t)dy
0
+
1
2
∫x
k(x − y, y)u(x − y, t)u(y, t)dy,
(149)
0
where γ is a constant to be determined and α is the index appearing in
(136). Then (Fγ , D(Fγ )) := (Fm,ω − γAα
m , D(Am )) generates an analytic
semigroup, say (SFγ (t))t≥0 , on Xm . Since (SFm,ω (t))t≥0 and (S−γAαm (t))t≥0 are
positive and contractive, we can use the Trotter product formula to deduce
that (SFγ (t))t≥0 is also a positive contraction on Xm . Furthermore, since
S−γAαm (t) ≤ Id for t ≥ 0, using again the Trotter formula
SFγ (t)u ≤ SFm,ω (t)u,
u ∈ Xm,+ .
(150)
(α)
and thus, for u ∈ Xm
2
(α)
∥SFγ (t)u∥(α)
m ≤ c1 ∥u∥m .
(151)
Next consider the set
(α)
U = {u ∈ Xm,+ : ∥u∥(α)
m ≤ 1 + b},
(152)
for some arbitrary fixed b > 0 and set
γ = 2K(b + 1).
Then on U we obtain
∫∞
(Cγ u)(x) := −u(x)
k(x, y)u(y)dy + (γ(aα
ω (x) + ω)u(x)
0
+
1
2
∫x
k(x − y, y)u(x − y)u(y)dy ≥ 0.
0
(153)
Kinetic models in natural sciences
Similarly, on U we have
∥Cγ u∥m ≤ K1 (U),
59
(154)
as well as
α
(α)
∥(γAα
m + ωI)u − (γAm + ωI)v∥m ≤ (ω + γ)∥u − v∥m
and
∥Cγ u − Cγ v∥m ≤ K2 (U)∥u − v∥(α)
m ,
◦
(155)
◦
(α)
(α)
for some constants K1 (U), K2 (U). Hence, for u ∈ Xm,+ satisfying ∥ u ∥m ≤
◦
c−2
1 b, for b of (152) and c1 from (151), there is τ = τ (u) such that the mapping
method for
∫t
◦
(T u)(t) = SFγ (t) u + SFγ (t − s)Cγ u(s)ds
0
is a contraction on Y = C([0, τ ], U), with U defined by (152) and the metric
◦
(α)
(α)
induced by the norm ∥u(t)∥Y := sup ∥u(t)∥m . Therefore, for any u ∈ Xm,+ ,
0≤t≤τ
(α)
there is a unique mild solution u to (144) in Xm,+ which, moreover, satisfies
u ∈ C 1 ((0, τ ), Xm ) ∩ C((0, τ ), D(Am )).
Proof of global solvability.
The local solution, constructed in the previous section, can be extended in
◦
a usual way to the maximal forward interval of existence [0, τmax (u)). Thus,
(α)
to show that u is globally defined, we need to show that ∥u(t)∥m is a priori
bounded uniformly in time.
Let us denote by Mr the r-th moment of u,
∫∞
xr u(x)dx.
Mr (u) :=
0
Then, for some constant L,
∫∞
∥u∥(α)
m
≤L
|u(x)|(1 + xm+jα )dx = L(M0 (u) + Mm+jα (u)).
(156)
0
Though for a given m, Theorem 23 does not ensure the differentiability of
Mm+αj , it is valid in the scale of spaces Xr with r ≥ m provided, of course,
◦
(α)
u ∈ Xr(α) . Since the embedding Xr(α) ⊂ Xm
is continuous for r ≥ m, the
60
Jacek Banasiak
◦
(α)
(α)
solutions emanating from the same initial value u ∈ Xr ⊂ Xm in each
◦
(α)
space, by construction, must coincide. Hence, let u ∈ Xm+jα ⊂ Xm+jα ⊂
(α)
Xm so that
◦
◦
(α)
◦
u ∈ C([0, τmax (u)), Xm+jα )∩C 1 ((0, τmax (u)), Xm+jα )∩C((0, τmax (u)), D(Am+jα )),
◦
with possibly different, but still nonzero, τmax (u). This, in particular, yields
differentiability of ∥u(·)∥0 = M0 (u(·)) and, consequently, of Mm+jα (u(·)). To
get the moment estimates we use the inequality
(x + y)r − xr − y r ≤ (2r − 1)(xr−1 y + y r−1 x) =: Gr (xr−1 y + y r−1 x), (157)
for r ≥ 1, x, y ∈ R+ . Then
∫∞
xr (Cu)(x)dx
0
= Gr KLα (Mr+jα−1 M1 + Mr−1 M1+jα + 2Mr−1 M1 ).
(158)
For the particular cases r = 0 and r = 1 we obtain
∫∞
∫∞
1
(Cu)(x)dx = −
k(x, y)u(x, t)u(y, t)dxdy ≤ 0,
2
0
0
∫∞
x(Cu)(x)dx = 0.
0
◦
Hence, using estimates for the linear part, we obtain on (0, τmax (u))
M0,t ≤ 4a0 b0 (M0 + Mm ),
M1,t = 0,
(159)
Mm+jα,t ≤ Gm+jα KLα (Mm+2jα−1 M1 + Mm+jα−1 (M1+jα + 2M1 )).
We see that if 1 ≤ r ≤ r′ , then
Mr ≤ M1 + Mr′
(160)
′
as xr ≤ x on [0, 1] and xr ≤ xr on [1, ∞). Thus, we see that in order for the
moment system (159) to be closed, we must assume that
jα ≤ 1.
This allows us to re-write (159) as
M0,t ≤ 4a0 b0 (2M0 + Mm+jα ),
Mm+jα,t ≤ Gm+jα KLα ((Mm+jα + M1 )M1
+(Mm+jα + M1 )(M2 + 3M1 )),
(161)
(162)
Kinetic models in natural sciences
61
where M1 is constant and where we used jα ≤ 1. To find the behaviour of
M2 , again we use (142) and (158), with an obvious simplification of (157), to
get the estimate for M2 as
M2,t ≤ 4KLα (M1+jα M1 + M12 ) ≤ 4KLα (M2 M1 + 2M12 ).
Hence, M2 is bounded on its interval of existence. Then, from the second inequality in (162), we see that Mm+jα satisfies a linear inequality with bounded
◦
coefficients and thus it also is bounded on (0, τmax (u)). This in turn yields the
(α)
boundedness of M0 . Hence, ∥u(·)∥m is bounded and thus u exists globally.
To ascertain global existence of solutions emanating from any initial datum
◦
(α)
(α)
(α)
u∈ Xm
we observe that since Xm+jα is dense in Xm , finite blow-up of such a
solution would contradict the theorem on continuous dependence of solutions
on the initial data.
⊓
⊔
Example 8. Suppose we have a(x) = xj , j > 0, and k(x, y) = xβ + y β . Then
we can write
k(x, y) = a(x)β/j + a(y)β/j
so that α = β/j. The assumption for local solvability require α < 1; that is
β < j while for the global solvability we additionally need αj ≤ 1; that is
β ≤ 1. On the other hand, if k(x, y) = xβ y β , then the conditions of the local
solvability remain the same while from
2xβ y β ≤ x2β + y 2β ,
it follows that we require β ≤ 1/2.
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