Hyperbolic problems with discontinuous coefficients

Linear and nonlinear hyperbolic PDEs arise in all sciences (physics, chemistry, medicine, engineering, astronomy, etc). In particular, in physics they model several important phenomena, from propagation of waves in a medium (for instance propagation of seismic waves during an earthquake) to refraction in crystals and gas-dynamics. When modelling wave propagation trough a multi-layered medium, for instance the subsoil during an earthquake, it is physically meaningful to make use of discontinuous functions.

This project wants to study the largest possible class of hyperbolic equations and systems: with variable multiplicities and discontinuous coefficients (depending on time and space). This is notoriously a very difficult problem due to the presence of multiplicities and the low-regularity of the coefficients. It will require the development of new analytical methods which will be first introduced under assumptions of regularity (first part of the project) and then gradually adapted to less regular coefficients (second part of the project).

In order to provide a unified approach to hyperbolic problems with discontinuous coefficients, we will test the strength of our new analytical methods numerically. This will build a bridge between two different approaches to hyperbolic PDEs (analytical and numerical), a bridge based on analysis, comparison and implementation of new ideas.

The results of the proposed research on hyperbolic problems with discontinuous coefficients will have direct impact on a wide range of scientific disciplines which employ hyperbolic equations in their mathematical models: seismology and geophysics (transmission of waves during earthquakes or tsunamis), engineering (acoustics and elastic waves), medical imagining and tomography, to mention a few.

Note that, the potential application of this project to seismology and geophysics (the propagation of waves in a layered medium as the Earth can be studied via a hyperbolic system with singular coefficients and possible multiplicities) has an important social impact as well. Indeed, everything which allows us to know more about the internal structure of our planet not only leads to important advances in Science but could help humanity to better deal with calamities like earthquakes and tsunamis.

Finally, this project will have a great mathematical impact on the academics working on related research subjects: PDEs, microlocal analysis, propagation of singularities, global analysis, functional analysis and numerical analysis.