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NEW DOCTORAL DEGREE Contribution to the theory of Friedrichs' and hyperbolic systems
Krešimir Burazin
orcid.org/0000-0001-6713-7560
; Department of Mathematics, University of Osijek, Osijek, Croatia
Abstract
We study various topics concerning Friedrichs' systems (first two chapters)
and first-order semilinear hyperbolic systems (third chapter).
In the first part the basic properties of graph spaces of first-order
differential operators are given, with special emphasis on investigation
of properties of the trace operator, which plays a major
role when imposing boundary conditions.
Here we follow Ph.D. thesis of M.\,Jensen
(2004), with
proofs being simplified and shortened.
In the second part we study Friedrichs' systems, a class of boundary value
problems that admit the study of a wide
range of differential equations in a unified framework.
They were introduced by K.~O.~Friedrichs
in 1958 ~as an attempt to treat equations of mixed type (such as Tricomi equation).
In a recent paper: {\sc A.~Ern, J.-L.~Guermond, G.~Caplain}: {\em An Intrinsic Criterion for the Bijectivity
of Hilbert Operators Related to Friedrichs' Systems}, Comunications in Partial
Differential Equations {\bf 32}(2007), 317--341;
a new view on the theory of Friedrichs systems was given,
as the theory is written
in terms of Hilbert spaces, and a new way of representation of
boundary conditions was introduced.
Here, the admissible boundary conditions are characterized by two intrinsic geometric conditions
in graph space, which avoids invoking traces at the boundary. Authors show that
these conditions imply maximality of boundary conditions.
They also introduce another representation of boun\-dary
conditions via a boundary operator, and show that this representation is equivalent with
intrinsic one (that enforced by two geometric conditions) if two specific operators $P$ and
$Q$ exist.
We have noted that these two geometric conditions can be naturally written in terminology
of an indefinite inner product on graph space, and use of classical results in Krein spaces
allowed us to construct the counter--example, which shows that operators
$P$ and $Q$ do not always exist. We also investigate situations when the existence of
$P$ and $Q$ is guarantied -- the case of one space dimension.
By using Kreine space we show that maximality of the boundary condition implies
intrinsic (geometric) conditions, and give a more elegant proof for the converse statement.
The relation between a {\sl classical} representation of admissible
boundary conditions (via matrix
fields on boundary), and those given by the boundary operator is addressed as well:
necessary conditions on the boundary matrix in order to define the boundary operator with
satisfactory properties are given, followed by some examples.
\newpage
The third part is concerned with first-order decoupled semilinear hyperbolic systems.
The local existence and uniqueness result can be found in {L.~Tartar}:
{\em From Hyperbolic Systems to Kinetic Theory, A Personalized Quest},
Sprin\-ger-Verlag, Berlin Heidelberg 2008, and
it is paired with an estimate on the solution of a certain type. We prove
it under slightly generalized assumptions that do
not change the proof, but allow a more precise estimate on the solution.
The estimates on
the solution and the time of its existence is the main topic of this chapter.
It is shown how to achieve the best possible estimate on the solution
and its time of existence (the best among all estimates of a certain type---the type
provided by the existence and uniqueness theorem).
The ${\rm L}^p$ version (for $1
existence and uniqueness theorem is also briefly discussed.
Keywords
Hrčak ID:
53706
URI
Publication date:
3.6.2009.
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