Hyperbolic problems with discontinuous coefficients
Lead Research Organisation:
Loughborough University
Department Name: Mathematical Sciences
Abstract
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.
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.
Planned Impact
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.
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.
Publications
Avetisyan Z
(2023)
A note on the polar decomposition in metric spaces
Discacciati M
(2023)
On the Wave Equation with Space Dependent Coefficients: Singularities and Lower Order Terms
in Acta Applicandae Mathematicae
Discacciati M
(2022)
Inhomogeneous wave equation with t-dependent singular coefficients
in Journal of Differential Equations
Garetto C
(2024)
Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, III: singular coefficients
in Mathematische Annalen
Ghosh S
(2023)
Compact embeddings, eigenvalue problems, and subelliptic Brezis-Nirenberg equations involving singularity on stratified Lie groups
in Mathematische Annalen
Kumar V
(2023)
Stein-Weiss inequality on non-compact symmetric spaces
Ruzhansky M
(2023)
Wave Equation for Sturm-Liouville Operator with Singular Intermediate Coefficient and Potential
in Bulletin of the Malaysian Mathematical Sciences Society
Description | This refers to EP/V005529/2 In this first year of this grant we have investigated mainly second order hyperbolic equations with multiplicities providing a deeper insight into very weak solutions. This has been done via numerical methods in a published paper and in a second one currently under review. |
Exploitation Route | Please refer to EP/V005529/2 We have presented our results to conferences and workshop and we are planning a future grant application inspired by this first results. |
Sectors | Other |
Description | This refers to EP/V005529/2 These first findings have inspired some current work on higher order hyperbolic equations and more in general on p-evolution equations. Two new research collaborations have been established. |
Description | Collaboration with Ferrara University |
Organisation | University of Ferrara |
Country | Italy |
Sector | Academic/University |
PI Contribution | Collaboration with Dr Alessia Ascanelli |
Collaborator Contribution | Research work on p-evolution equations with Cingular coefficients |
Impact | Work in progress |
Start Year | 2023 |
Description | Collaboration with Torino University |
Organisation | University of Turin |
Country | Italy |
Sector | Academic/University |
PI Contribution | Collaboration with Dr Marco Cappiello and Dr Alexandre Arias Junior to work on p-evolution equations with singular coefficients |
Collaborator Contribution | Research contribution |
Impact | Work in progress |
Start Year | 2023 |
Description | Online Workshop on hyperbolic equations |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Other audiences |
Results and Impact | Online workshop with a diverse audience in terms of nationality, gender and career stages. |
Year(s) Of Engagement Activity | 2021 |
URL | https://ruzhansky.org/lms-hyperbolic-network/ |