#### Eric Hester

- Asymptotic analysis of PDEs
- Spectral algorithms for numerical PDE solvers
- Multiphase fluid mechanics

Department of Mathematical Sciences

**Dr Matthew Schrecker**

Much of my research to date has focused on the compressible Euler equations. The Euler equations have a rich mathematical structure underpinning one of the most basic physical processes in the universe: the flow of gases. Over the past 250 years, the equations have inspired developments in mathematical analysis, functional and harmonic analysis, dynamical systems and geometric analysis. The Euler equations are also relevant in understanding phenomena in the physical world ranging from the flow out of an exhaust pipe to the motion of galaxies and collapse of stars. In all of these physical processes, the formation of shock waves is a ubiquitous phenomenon (as seen physically, for example, in a sonic boom coming from a Concorde aircraft) that causes a drastic loss of regularity in solutions of the system. The presence of shocks has led to the need to study weak solutions of the Euler equations, as classical (differentiable) solutions will generically blow up in finite time, and so my research has focused on both of these aspects: the existence of weak solutions and the properties and behaviour of smooth solutions on approach to singularity formation.

More recently, I have also been working on the problem of the gravitational collapse of stars, a different type of singularity formation in which the physical variables of the system (particularly the density of a star) blow up in finite time as matter falls in towards the centre of the star. Such a phenomenon is also modelled with the Euler equations, but now with a gravitational field as well.

**Research Interests:**

- Analysis of Partial Differential Equations
- Fluid dynamics (especially free boundary problems and nonlinear singularity formation)
- Shock waves (their formation, structure and dynamics)

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