Moving boundary problems in the field of PDEs, specifically the relativistic Euler problem.
Lead Research Organisation:
King's College London
Department Name: Mathematics
Abstract
Trying to show global stability for small enough data for the relativistic Euler equations, with an expanding metric. Methodology is to use the expanding metric to infer exponential decay on certain norms which will then ensure the energy is bounded for all time.
Organisations
People |
ORCID iD |
| Mahir Hadzic (Primary Supervisor) | |
| Shrish Parmeshwar (Student) |
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509498/1 | 01/10/2016 | 30/09/2021 | |||
| 1948162 | Studentship | EP/N509498/1 | 01/01/2017 | 30/09/2020 | Shrish Parmeshwar |
| Description | We started out investigating the Compressible Free Boundary (Special) Relativistic Euler Equations, which describe a fluid moving in a fixed relativistic spacetime. Finding solutions to these equations in expanding space-times proved difficult for technical, mathematical reasons. We then looked at a related problem for fluids that can be described more classically, and asked whether one could find solutions to the Compressible Free Boundary Euler Equations, without any relativity, in which the fluid is still expanding. Note that this had its own difficulties due to the fact that in the special relativistic case, the expanding space-time automatically meant the fluid would look to expand, where as in the classical case, our space-time had no expansion, and so mathematically this mean we had to force the fluid to expand. Nevertheless this proved more successful. In the end we showed: One can find expanding solutions to the Compressible Free Boundary Euler Equations that expand, by taking advantage of scaling properties inherent to the equations. This is significant as previously such solutions had only been found by reducing the problem to a finite dimensional one, a reduction that was not needed in our method. We also showed this result for the Compressible Free Boundary Euler-Poisson Equations, which model a fluid coupled with gravity, and is a classical model for a Newtonian Star, without any relativistic effects. Finally we found expanding solutions to the N-Body Compressible Free Boundary Euler-Poisson Equations, which describe multiple stars interacting with each other via gravity. This was significant as gravitational forces attract stars to one another, and thus we had to find initial velocities and positions for each star such that these forces would not cause the stars to move towards each other. |
| Exploitation Route | In terms of research directions, one can look to produce the same sort of results for the corresponding Navier-Stokes system, which has additional dispersive effects, and one would expect these would only help finding expanding solutions, rather than hinder it. There is also the opportunity to try and go back to the relativistic problem, and address the technical issues which made it difficult to make progress, compared to the classical problem. |
| Sectors | Other |
| URL | https://arxiv.org/search/math?searchtype=author&query=Parmeshwar%2C+S |