Conormal distributions in nonlinear hyperbolic problems

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 space dimensions.

Historical remark

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.