Unit:
Κατεύθυνση Εφαρμοσμένα ΜαθηματικάLibrary of the School of Science
Author:
Αβαράκης Δημήτριος
Supervisors info:
Ιωάννης Στρατής Καθηγητής (Επιβλέπων), Γεράσιμος Μπαρμπάτης Αναπλ. Καθηγ., Χριστόδουλος Αθανασιάδης Καθηγητής
Original Title:
Eπιλύουσες οικογένειες τελεστών για αυτόνομα συστήματα σε απειροδιάστατους χώρους
Translated title:
Pesolving families of operators for autonomous systems in infinite-dimensional spaces
Summary:
The purpose of this M.Sc. thesis is to present the well posedness of the
abstract
Cauchy problem in infinite dimensional Banach spaces, as well as some
applications
to pde’s. A necessary tool for the study of the abstract Cauchy problem is the
theory
of C0 semigroup of bounded linear operators in Banach spaces. In the first
chapter
we give some definitions from linear operator’s theory and some basic notions
of calculus of functions in Banach spaces. In the second chapter we study the
Hille - Yosida theorem and some results induced by this and we give necessary
and sufficient conditions for the generation of C0 semigroups. Furthermore, we
study the analytic semigroups associated with sectorial operators and their
strong
properties as a particular class of strongly continuous semigroups. Next, we
give
the notions of classical, strong and mild solution of the autonomous
non-homogeneous
Cauchy problem and we present the conditions for the (Hadamard) well posedness
of the problem. In the third chapter we study some applications arising in
physical
problems, specifically in electromagnetism and heat transfer. Considering these
problems as abstract Cauchy problems in suitable Banach spaces, we apply the
theory that we have developed to show the well posedness of these problems.
Keywords:
Cauchy problem, Analytic semigroups, Strongly continuous semigroups, Maxwell operator, Resolvent operator
