RESEARCH PROJECT
Title: generalized solutions to quasilinear hyperbolic systems
The research project addresses quasilinear hyperbolic systems of partial differential
equations (PDEs), more specifically, systems of conservation laws.
In recent years, new approaches to this important problem have been developed both
in pure and applied PDE's. In particular, it became clear that in order to solve certain
types of systems, new types of generalized solutions were needed: so called delta and
singular shock waves [1, 2]. These new solutions have singularities containing delta
functions or some rational power of thereof. Because of this fact, the usual
Approaches in this field fail and a new notion of generalized solution is needed.
The main tool to be used here will be the theory of Colombeau generalized functions.
The proposed project could be described as follows:
The first step should be the construction of generalized solutions to the so-called
Riemann problem (piecewise constant initial data) in an appropriate space.
The second step will be to give a classical interpretation of the obtained solutions.
The third step will be the selection of the physically relevant solutions ("admisibility"
of solutions).
Questions of this type have been addressed by Prof. Marko Nedeljkov (Novi Sad) and
Prof. Michael Oberguggenberger (Innsbruck) during the research visits of the former
in Innsbruck under the OAD-projects 1778-4/1997 and 1236-2/2000 during a total
period of eleven months in 1997/1998 and 2000/2001, see [3]. While I have already
contributed in a joint paper with Prof. Nedeljkov to the subject [4], I am now at the
beginning of my master's studies in Novi Sad. My principal advisor will be Prof.
Nedeljkov; it is planned that Prof. Oberguggenberger will act as an external coadvisor. For this reason, it will be important for me that I can establish the basis for
my thesis work during the proposed visit to Innsbruck and in this way
Prof.Oberguggenberger can be involved in my research right from the beginning.
A stay of mine in Innsbruck during a full semester's term of four months (intended
for the summer semester 2004) will give me the opportunity to participate in the
analysis seminars of the research group around Prof. Oberguggenberger at the
Department of Engineering Mathematics, Geometry and Computer Science in
Innsbruck.
DETAILS THE PLANNED RESEARCH:
A very important question, the uniqueness of solutions as discussed above,is a vast
area. This problem will be approached during the project. It should at least be possible
to set some limits to non-uniqueness.
Generaly speaking, for solving arbitrary Cauchy problems, the problem of interactions
of the above type of solutions with other waves arises. This question will also be
under consideration in this project.
A further aim will be to collect and organize known facts aboutgeneralized solutions
to quasilinear hyperbolic PDEs in order to develop a concise theory suitable for
teaching and for applications in other physical sciences.
Jelena Pasic
REFERENCES
[1] Keyfitz, B. L. and Kranzer, H. C.: Spaces of weighted measures for conservation
laws with singular shock solutions. J. Diff. Eqs. 118(1995), 420 - 451.
[2] M. Nedeljkov: Delta and singular delta locus for one dimensional systems of
conservation laws. Math. Mod. Appl. Sci., to appear.
[3] M. Nedeljkov, M. Oberguggenberger: Delta shock wave and interactions in
a model case. Preprint, 2002.
[4] M. Nedeljkov, J. Pasic: A note on singular shock wave interactions. Preprint,
2002.
SHORT DESCRIPTION
The research project addresses quasilinear hyperbolic systems of partial differential
equations (PDEs), more specifically, systems of conservation laws. The main tool to
be used here will be the theory of Colombeau generalized functions. The proposed
project could be described as follows: The first step should be the construction of
generalized solutions to the Riemann problem in an appropriate space. The second
step will be to give a classical interpretation of the obtained solutions. The third step
will be the selection of the physically relevant solutions.Further, the uniqueness
problem and the interaction of such solutions with other waves will be investigated.