Microlocal analysis on algebras of generalized functions

The project presents a systematic extension of microlocal analysis and regularity theory to partial differential operators with generalized functions as coefficients. In applications, equations involving such operators arise, for example, in models of wave propagation in highly irregular media (such as seismic waves in the earth`s subsurface). The typical source mechanisms are strong pulses (e.g. Dirac deltas), which enter as highly singular initial data or right-hand sides in the model equations. The corresponding propagating singular (non-smooth) disturbances in the wave solution represent the leading order information about the medium properties (and the source). Mathematically, this requires an analysis of propagation of singularities of generalized solutions to partial differential equations, whose coefficients are also singular (often even discontinuous). Thus nonlinear interactions of singularities arise, which are beyond the reach of analysis in classical function and distribution spaces. The aim of the project is to establish a comprehensive theory of microlocal analysis, i.e., spectral analysis of singularities, in algebras of generalized functions. It will be based on a systematic development of a new generalized pseudodifferential calculus and a new theory of oscillatory integrals with singular phase functions and amplitudes. For example, in a geometric optics-type construction (Fourier integral operator parametrix) of approximate solutions to hyperbolic equations the generalized phases and amplitudes reflect the qualitative medium properties. Thus it is crucial to have a thorough understanding of their microlocal influence on propagating initial value singularities. Furthermore, Fourier integral operators are a tool to transform the original problem geometrically into a standard situation, where propagation of singularities along the bicharacteristic flow can be directly analyzed. We will address the extension of such a correspondence to the case of a (non-smooth) generalized bicharacteristic flow.

Partial differential equations are one of the major modelling tools in science and engineering, describing physical quantities that vary in space and time. A typical example is wave propagation in liquids, gases and solids. The coefficients in these equations encode the material properties of the underlying medium in which such propagation takes place. The so-called initial data describe the physical mechanisms that produce the propagating disturbances. Ever more frequently modelling of such phenomena requires the incorporation of coefficients and data that are not usual smooth functions. An example is provided by seismic waves in the earth`s subsurface. Ruptures and discontinuities in the medium enter into the coefficients, and the measured waves are often initiated by strong pulses, such as earthquakes or explosions. Classical differential calculus cannot handle the simultaneous occurrence of strong irregularities in the coefficients and the data. About two decades ago, a new theory of generalized functions was initiated in which such differential equations with irregular coefficients could be solved in full generality. The present project addresses the properties of these generalized solutions, in particular, the location of discontinuities in the propagating waves. For the qualitative investigation of classical smooth solutions, rather high-level mathematical tools had been developed over the last half century. These tools form the bulk of knowledge now called microlocal analysis, that is the spectral theory of singularities, and make use of such advanced mathematical concepts as pseudodifferential operators and Fourier integral operators. The aim of this project was to develop an analogous theory, but for generalized solutions belonging to the wider class addressed above. The project succeeded in developing a complete theory of these Fourier integral operators in the generalized framework. The regularity properties of the solutions to equations with irregular coefficients can now be understood and described. In developing these tools, the project laid the foundations for future applications and a better qualitative understanding of what the physical models say about singularity propagation. In addition, a number of foundational results were obtained that contributed to progress in the theory of generalized functions itself. In particular, the classical concept of a distance between two functions (topology of function spaces) could be transferred to generalized functions with all its important theoretical consequences that can be inferred from it.