Supervisor: Dr Claudia Garetto
Project description:
The mathematical model of wave propagation can involve different types of partial differential equations (PDEs) depending on the physical context that one wants to study. For instance, propagation of acoustic waves or seismic waves in a multi-layered medium can be investigated via a system of first order strictly hyperbolic equations, provided that the velocity is positive. The wave function of a quantum-mechanical system is governed by the Schrödinger equation and the linearisation of several dispersive equations as KdV and KdV-Burgers appear in wave propagation in shallow water or in a liquid-filled elastic tube.
The aim of this project is to study a rather flexible class of partial differential equations which contains all the examples mentioned above: the so-called p-evolution equations, where p≥1. Note that strictly hyperbolic equations are obtained for p=1, Schrödinger type equations for p=2 and linearisations of KdV and KdV-Burger equations for p=3. The well-posedness of the corresponding Cauchy problem is widely understood when the equation coefficients are regular (at least continuous) but it is still an open problem when the coefficients are low regular and often when the space dimension is bigger than 1. In this project our aim is to employ suitable regularisation techniques to study p-evolution equations with low regular time-dependent coefficients. Solutions will be identified in a weak sense (very weak solutions) and the limiting behaviour will be investigated via theoretical and numerical methods.
This is a natural continuation of the research work of the second supervisor and I currently funded by the EPSRC grant EP/V005529/2: Hyperbolic problems with discontinuous coefficients (2021-2025). Note that I have previously investigated the case p=1 for low regular coefficients so the project will focus mainly on the cases p=2 and p=3 in space dimension 1.
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