Many hyperbolic equations in
mathematical physics are quasilinear, but have the benefit of coming in
the form of a conservation law. Notably the equations from continuum
mechanics and magnetohydrodynamics belong to this class. There is a
well-established theory of hyperbolic conservation laws in one space
dimension. In higher space dimensions, however, the situation is more
complicated, and it is one of the most pressing problems in the modern
theory of partial differential equations to settle a number of open
questions around this.
The difficulties stem from the fact
that solutions develop singularities in finite time. In one space
dimension, functions of bounded variation is the appropriate class of
functions to work with. This is know to fail in higher space
dimensions. So what one is looking for is a class of bounded functions
so narrow to exhibit nice multiplicative properties, but wide enough to
allow for singularities.
The appropriate class seems to be
conormal distributions (generalized functions) which are bounded. As a
matter of fact, conormal singularities is precisely the kind of
singularity (shocks, rarefaction waves, etc.) formed by solutions in
one space dimension. What is more, this kind of singularity has also
been observed in all known examples in higher space dimensions. From a
mathematical point of view, the class of conormal distributions is very
interesting, as these distributions admit local representations via
certain Fourier integrals, where the representation reflects the
geometric situation under consideration (keywords: resolution of
singularities, second microlocalization). This then makes techniques
from harmonic analysis and from nonlinear microlocal analysis
applicable which yields as a consequence the best results one can currently hope for.
This is basically the line of attack
we pursue in Göttingen. Namely, we are contributing to the completion of the theory
of conormal distributions putting much emphasis on nonlinear aspects.
The whole approach looks promising, but there is still a lot of
ground to cover before one can hope to reach a state what could be
called a satisfactory theory of hyperbolic conservation laws in higher
In gas dynamics, the theory of non-smooth solutions was initiated by
the ingenious Göttingen mathematician Bernhard Riemann in his
paper "Ueber die Fortpflanzung ebener Luftwellen von endlicher
Schwingungsweite", Abhandlungen der Königlichen Gesellschaft der
Wissenschaften zu Göttingen, 8 (1860), 43-65. Concepts like
Riemann problem, Riemann solver, etc. bear nowadays his name.