Project A13 • Dispersive estimates for wave equations with low regularity coefficients

Principal investigators

  Prof. Dr. Dorothee Frey (7/2021 - )
  Prof. Dr. Roland Schnaubelt (7/2021 - )

Project summary

Wave equations are ubiquitous, and the consideration of wave propagation in e.g. inhomogeneous media, domains with boundary, or in the modelling of nonlinear wave phenomena often requires the study of wave equations with low regularity coefficients, in particular coefficients with regularity less than $C^{1,1}$. Yet the mathematical investigation of their qualitative behaviour remains highly challenging, as the low regularity of the coefficients partly impedes the application of classical analytical methods such as the Fourier transform and Fourier restriction theory. In the case of coefficients that are at least $C^{1,1}$ regular, basic properties of both linear and nonlinear wave equations are by now well understood. For coefficients with regularity less than $C^{1,1}$, however, there exist counterexamples which show that the wave equation does not possess the same qualitative behaviour in general.

In this project we investigate low regularity wave equations for specific types of wave propagation. Even though the existing results on wave equations with $C^s$ coefficients with $s<2$ are often sharp in general, preliminary work shows that there exist specific classes of coefficients for which the available regularity results and Strichartz estimates can be considerably improved. This is in particular true when wave propagations in different directions are independent of each other.

We therefore put forward a new approach to Strichartz and dispersive estimates that exploits structural properties of the operators involved rather than solely relying on regularity properties of the coefficients. Our new approach is based on sophisticated coefficient-adapted localisation arguments and generalised Fourier integral operators. Since the main techniques we use are operator-theoretic, the approach is flexible enough to be applicable also in other low regularity settings that exhibit specific structural properties. We will thus also investigate wave equations on rough domains or other rough geometric settings. In the long-term perspective we intend to tackle Maxwell systems in collaboration with project A5, and explore applications to magnetic Schrödinger equations under additional commutator relations on the magnetic potential. The obtained results shall be applied to the wellposedness theory of nonlinear wave equations.

Our research proposal also includes the development of a comprehensive theory of hyperbolic Hardy spaces that are preserved under (generalised) Fourier integral operators and have the potential to serve as efficient solution spaces for iterative arguments in the context of nonlinear wave equations. This is in analogy to adapted parabolic Hardy spaces, which in recent years have become an indispensable tool in the theory of elliptic boundary value problems with $L^\infty$ coefficients.

Preprints

  1. and . $L^p$ estimates for wave equations with specific $C^{0,1}$ coefficients. CRC 1173 Preprint 2020/29, Karlsruhe Institute of Technology, October . Revised version from April 2021. [bibtex]