Hyperbolic systems with generalized functions as coefficients: generalized diffraction of waves

Research project P 14576 Hyperbolic systems with non-smooth coefficients Michael OBERGUGGENBERGER 09.10.2000 The project addresses mathematical problems of wave propagation and diffraction in discontinuous media with applications to geophysics. In direct and inverse problems from seismology and geophysics, propagation of acoustic waves is described by linear hyperbolic equations and systems. Due to material properties of the earth, the singularity structure of the coefficients is important. This encompasses jump discontinuities, fractal-like boundaries, or an irregular stochastic structure. On the other hand, seismic experiments involve source terms of delta-function type. Mathematically, the problem thus requires methods going beyond the classical schemes and allowing multiplication of distributions (be it microlocal methods or generalized function algebras). The goal of the project is to develop the mathematical tools which allow a consistent investigation of generalized solutions to linear hyperbolic equations with non-smooth coefficients to address the geophysical questions. The general framework will be the Colombeau theory of generalized functions. More specifically, the research will focus on existence and uniqueness theorems, qualitative studies, development of microlocal methods (wave front sets, wave splitting) and the study of propagation of singularities in this setting. The latter topic is important for the detection of material ruptures. The investigation is primarily mathematical in nature, but will be undertaken in close collaboration with the geophysics group around M. de Hoop at the Center of Wave Phenomena in Colorado, and will thus have an impact on geophysical research as well.

The project developed regularity theory for partial differential equations with non-smooth, generalized coefficients as well as applications to geophysical model equations of seismic wave propagation. Due to the interaction of singularities in the coefficients with a prospective solution classical approaches were known to be insufficient, in general, for various reasons: model equations are often not well-defined, non-existence and non-uniqueness of distributional solutions, lack of generally applicable solution concepts and methods of qualitative analysis. Most of our new methods and results are set in the framework of nonlinear extensions of distribution theory, namely Colombeau algebras of generalized functions, but we could also prove certain solvability and regularity results within classical function spaces. On the level of the general theory, we refined classical regularity notions and developed a number of conditions which imply preservation of regularity in the solution, either from source terms or initial data. The latter refers to the general so-called propagation of singularities. New pseudodifferential techniques were developed for this purpose and a new pseudodifferential characterization of generalized wave front sets was obtained. One special type of regularity investigated in detail was generalized Hölder-continuity. The corresponding result is an important contribution to recent research in geophysics which aims at precisely distinguishing material properties in geological units (e.g. sound speed variation). Here, our results are concerned with solvability in Sobolev-type function spaces on the one hand, and estimation of the wave solution regularity from that of the coefficients (i.e., material properties).