Mathematical Analysis and Numerical Methods for Science and Technology: Volume 5 Evolution Problems I
Autor Robert Dautray Traducere de A. Craig Contribuţii de M. Artola, M. Cessenat Autor Jacques Louis Lions Contribuţii de H. Lanchonen Limba Engleză Paperback – 22 noi 1999
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Specificații
ISBN-13: 9783540661016
ISBN-10: 3540661018
Pagini: 756
Ilustrații: 1
Dimensiuni: 155 x 235 x 40 mm
Greutate: 1.06 kg
Ediția:2000
Editura: Springer Berlin, Heidelberg
Colecția Springer
Locul publicării:Berlin, Heidelberg, Germany
ISBN-10: 3540661018
Pagini: 756
Ilustrații: 1
Dimensiuni: 155 x 235 x 40 mm
Greutate: 1.06 kg
Ediția:2000
Editura: Springer Berlin, Heidelberg
Colecția Springer
Locul publicării:Berlin, Heidelberg, Germany
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Public țintă
ResearchDescriere
299
G(t),
and
to
obtain
the
corresponding
properties
of
its
Laplace
transform
(called
the
resolvent
of
-
A)
R(p)
=
(A
+
pl)-l
,
whose
existence
is
linked
with
the
spectrum
of
A.
The
functional
space
framework
used
will
be,
for
simplicity,
a
Banach
space(3).
To
summarise,
we
wish
to
extend
definition
(2)
for
bounded
operators
A,
i.e.
G(t)
=
exp(
-
tA)
,
to
unbounded
operators
A
over
X,
where
X
is
now
a
Banach
space.
Plan
of
the
Chapter
We
shall
see
in
this
chapter
that
this
enterprise
is
possible,
that
it
gives
us
in
addition
to
what
is
demanded
above,
some
supplementary
information
in
a
number
of
areas:
-
a
new
'explicit'
expression
of
the
solution;
-
the
regularity
of
the
solution
taking
into
account
some
conditions
on
the
given
data
(u
,
u1,f
etc
...
)
with
the
notion
of
a
strong
solution;
o
-
asymptotic
properties
of
the
solutions.
In
order
to
treat
these
problems
we
go
through
the
following
stages:
in
§
1,
we
shall
study
the
principal
properties
of
operators
of
semigroups
{G(t)}
acting
in
the
space
X,
particularly
the
existence
of
an
upper
exponential
bound
(in
t)
of
the
norm
of
G(t).
In
§2,
we
shall
study
the
functions
u
E
X
for
which
t
--+
G(t)u
is
differentiable.
Cuprins
XIV.
Evolution
Problems:
Cauchy
Problems
in
IRn.-
§1.
The
Ordinary
Cauchy
Problems
in
Finite
Dimensional
Spaces.-
1.
Linear
Systems
with
Constant
Coefficients.-
2.
Linear
Systems
with
Non
Constant
Coefficients.-
§2.
Diffusion
Equations.-
1.
Setting
of
Problem.-
2.
The
Method
of
the
Fourier
Transform.-
3.
The
Elementary
Solution
of
the
Heat
Equation.-
4.
Mathematical
Properties
of
the
Elementary
Solution
and
the
Semigroup
Associated
with
the
Heat
Operator.-
§3.
Wave
Equations.-
1.
Model
Problem:
The
Wave
Equation
in
?n.-
2.
The
Euler—Poisson—Darboux
Equation.-
3.
An
Application
of
§2
and
3:
Viscoelasticity.-
§4.
The
Cauchy
Problem
for
the
Schrödinger
Equation,
Introduction.-
1.
Model
Problem
1.
The
Case
of
Zero
Potential.-
2.
Model
Problem
2.
The
Case
of
a
Harmonic
Oscillator.-
§5.
The
Cauchy
Problem
for
Evolution
Equations
Related
to
Convolution
Products.-
1.
Setting
of
Problem.-
2.
The
Method
of
the
Fourier
Transform.-
3.
The
Dirac
Equation
for
a
Free
Particle.-
§6.
An
Abstract
Cauchy
Problem.
Ovsyannikov’s
Theorem.-
Review
of
Chapter
XIV.-
XV.
Evolution
Problems:
The
Method
of
Diagonalisation.-
§1.
The
Fourier
Method
or
the
Method
of
Diagonalisation.-
1.
The
Case
of
the
Space
?1(n
=
1).-
2.
The
Case
of
Space
Dimension
n
=
2.-
3.
The
Case
of
Arbitrary
Dimension
n.-
Review.-
§2.
Variations.
The
Method
of
Diagonalisation
for
an
Operator
Having
Continuous
Spectrum.-
1.
Review
of
Self-Adjoint
Operators
in
Hilbert
Spaces.-
2.
General
Formulation
of
the
Problem.-
3.
A
Simple
Example
of
the
Problem
with
Continuous
Spectrum.-
§3.
Examples
of
Application:
The
Diffusion
Equation.-
1.
Example
of
Application
1:
The
Monokinetic
Diffusion
Equation
for
Neutrons.-
2.
Example
of
Application
2:
The
Age
Equation
in
Problems
of
Slowing
Down
of
Neutrons.-
3.
Example
of
Application
3:
Heat
Conduction.-
§4.
The
Wave
Equation:
Mathematical
Examples
and
Examples
of
Application.-
1.
The
Case
of
Dimension
n
=
1.-
2.
The
Case
of
Arbitrary
Dimension
n.-
3.
Examples
of
Applications
for
n
=
1.-
4.
Examples
of
Applications
for
n
=
2.
Vibrating
Membranes.-
5.
Application
to
Elasticity;
the
Dynamics
of
Thin
Homogeneous
Beams.-
§5.
The
Schrödinger
Equation.-
1.
The
Cauchy
Problem
for
the
Schrödinger
Equation
in
a
Domain
?
=
]0,
1[?
?.-
2.
A
Harmonic
Oscillator.-
Review.-
§6.
Application
with
an
Operator
Having
a
Continuous
Spectrum:
Example.-
Review
of
Chapter
XV.-
Appendix.
Return
to
the
Problem
of
Vibrating
Strings.-
XVI.
Evolution
Problems:
The
Method
of
the
Laplace
Transform.-
§1.
Laplace
Transform
of
Distributions.-
1.
Study
of
the
Set
If
and
Definition
of
the
Laplace
Transform.-
2.
Properties
of
the
Laplace
Transform.-
3.
Characterisation
of
Laplace
Transforms
of
Distributions
of
L+
(?).-
§2.
Laplace
Transform
of
Vector-valued
Distributions.-
1.
Distributions
with
Vector-valued
Values.-
2.
Fourier
and
Laplace
Transforms
of
Vector-valued
Distributions.-
§3.
Applications
to
First
Order
Evolution
Problems.-
1.
‘Vector-valued
Distribution’
Solutions
of
an
Evolution
Equation
of
First
Order
in
t.-
2.
The
Method
of
Transposition.-
3.
Application
to
First
Order
Evolution
Equations.
The
Hilbert
Space
Case.
L2
Solutions
in
Hilbert
Space.-
4.
The
Case
where
A
is
Defined
by
a
Sesquilinear
Form
a(u,
v).-
§4.
Evolution
Problems
of
Second
Order
in
t.-
1.
Direct
Method.-
2.
Use
of
Symbolic
Calculus.-
Review.-
§5.
Applications.-
1.
Hydrodynamical
Problems.-
2.
A
Problem
of
the
Kinetics
of
Neutron
Diffusion.-
3.
Problems
of
Diffusion
of
an
Electromagnetic
Wave.-
4.
Problems
of
Wave
Propagation.-
5.
Viscoelastic
Problems.-
6.
A
Problem
Related
to
the
Schrödinger
Equation.-
7.
A
Problem
Related
to
Causality,
Analyticity
and
Dispersion
Relations.-
8.
Remark
10.-
Review
of
Chapter
XVI.-
XVII.
Evolution
Problems:
The
Method
of
Semigroups.-
A.
Study
of
Semigroups.-
§1.
Definitions
and
Properties
of
Semigroups
Acting
in
a
Banach
Space.-
1.
Definition
of
a
Semigroup
of
Class
&0
(Resp.
of
a
Group).-
2.
Basic
Properties
of
Semigroups
of
Class
&0.-
§2.
The
Infinitesimal
Generator
of
a
Semigroup.-
1.
Examples.-
2.
The
Infinitesimal
Generator
of
a
Semigroup
of
Class
&0.-
§3.
The
Hille—Yosida
Theorem.-
1.
A
Necessary
Condition.-
2.
The
Hille—Yosida
Theorem.-
3.
Examples
of
Application
of
the
Hille—Yosida
Theorem.-
§4.
The
Case
of
Groups
of
Class
&0
and
Stone’s
Theorem.-
1.
The
Characterisation
of
the
Infinitesimal
Generator
of
a
Group
of
Class
&0.-
2.
Unitary
Groups
of
Class
&0.
Stone’s
Theorem.-
3.
Applications
of
Stone’s
Theorem.-
4.
Conservative
Operators
and
Isometric
Semigroups
in
Hilbert
Space.-
Review.-
§5.
Differentiable
Semigroups.-
§6.
Holomorphic
Semigroups.-
§7.
Compact
Semigroups.-
1.
Definition
and
Principal
Properties.-
2.
Characterisation
of
Compact
Semigroups.-
3.
Examples
of
Compact
Semigroups.-
B.
Cauchy
Problems
and
Semigroups.-
§1.
Cauchy
Problems.-
§2.
Asymptotic
Behaviour
of
Solutions
as
t
?
+
?.
Conservation
and
Dissipation
in
Evolution
Equations.-
§3.
Semigroups
and
Diffusion
Problems.-
§4.
Groups
and
Evolution
Equations.-
1.
Wave
Problems.-
2.
Schrödinger
Type
Problems.-
3.
Weak
Asymptotic
Behaviour,
for
t
?
±
?,
of
Solutions
of
Wave
Type
of
Schrödinger
Type
Problems.-
4.
The
Cauchy
Problem
for
Maxwell’s
Equations
in
an
Open
Set
?
?
?3.-
§5.
Evolution
Operators
in
Quantum
Physics.
The
Liouville—von
Neumann
Equation.-
1.
Existence
and
Uniqueness
of
the
Solution
of
the
Cauchy
Problem
for
the
Liouville—von
Neumann
Equation
in
the
Space
of
Trace
Operators.-
2.
The
Evolution
Equation
of
(Bounded)
Observables
in
the
Heisenberg
Representation.-
3.
Spectrum
and
Resolvent
of
the
Operator
h.-
§6.
Trotter’s
Approximation
Theorem.-
1.
Convergence
of
Semigroups.-
2.
General
Representation
Theorem.-
Summary
of
Chapter
XVII.-
XVIII.
Evolution
Problems:
Variational
Methods.-
Orientation.-
§1.
Some
Elements
of
Functional
Analysis.-
1.
Review
of
Vector-valued
Distributions.-
2.
The
Space
W(a,
b;
V,
V’).-
3.
The
Spaces
W(a,
b;
X,
Y).-
4.
Extension
to
Banach
Space
Framework.-
5.
An
Intermediate
Derivatives
Theorem.-
6.
Bidual.
Reflexivity.
Weak
Convergence
and
Weak
*
Convergence.-
§2.
Galerkin
Approximation
of
a
Hilbert
Space.-
1.
Definition.-
2.
Examples.-
3.
The
Outline
of
a
Galerkin
Method.-
§3.
Evolution
Problems
of
First
Order
in
t.-
1.
Formulation
of
Problem
(P).-
2.
Uniqueness
of
the
Solution
of
Problem
(P).-
3.
Existence
of
a
Solution
of
Problem
(P).-
4.
Continuity
with
Respect
to
the
Data.-
5.
Appendix:
Various
Extensions
—
Liftings.-
§4.
Problems
of
First
Order
in
t
(Examples).-
1.
Mathematical
Example
1.
Dirichlet
Boundary
Conditions.-
2.
Mathematical
Example
2.
Neumann
Boundary
Conditions.-
3.
Mathematical
Example
3.
Mixed
Dirichlet—Neumann
Boundary
Conditions.-
4.
Mathematical
Example
4.
Bilinear
Form
Depending
on
Time
t.-
5.
Evolution,
Positivity
and
‘Maximum’
of
Solutions
of
Diffusion
Equations
in
Lp(?),
1
?
p
?
?.-
6.
Mathematical
Example
5.
A
Problem
of
Oblique
Derivatives.-
7.
Example
of
Application.
The
Neutron
Diffusion
Equation.-
8.
A
Stability
Result.-
§5.
Evolution
Problems
of
Second
Order
in
t.-
1.
General
Formulation
of
Problem
(P1).-
2.
Uniqueness
in
Problem
(P1).-
3.
Existence
of
a
Solution
of
Problem
(P1).-
4.
Continuity
with
Respect
to
the
Data.-
5.
Formulation
of
Problem
(P2).-
§6.
Problems
of
Second
Order
in
t.
Examples.-
1.
Mathematical
Example
1.-
2.
Mathematical
Example
2.-
3.
Mathematical
Example
3.-
4.
Mathematical
Example
4.-
5.
Application
Examples.-
§7.
Other
Types
of
Equation.-
1.
Schrödinger
Type
Equations.-
2.
Evolution
Equations
with
Delay.-
3.
Some
Integro-Differential
Equations.-
4.
Optimal
Control
and
Problems
where
the
Unknowns
are
Operators.-
5.
The
Problem
of
Coupled
Parabolic-Hyperbolic
Transmission.-
6.
The
Method
of
‘Extension
with
Respect
to
a
Parameter’.-
Review
of
Chapter
XVIII.-
Table
of
Notations.-
of
Volumes
1–4,
6.