Timothy Collier is a postgraduate student at the University of Sydney who has recently completed a Bachelor of Science (Adv. Maths) majoring in Mathematics and Physics. Timothy has previously been involved in projects at the University of Sydney in harmonic function theory and the perturbation theory of eigenvalues as well as research in quantum error correction and quantum Hall effect circulators with Prof. Andrew Doherty of the University’s quantum science research group. He intends to continue with Honours in Mathematics, focusing on the Navier-Stokes equations under the supervision of Dr. Daniel Hauer.
The Time-dependent De Georgi Method for Operators
Consider a container filled with some fluid like water or oil. Every fluid admits a viscosity depending on its physical properties. For a given initial state and external force, the motion of the fluid in the container is described by the celebrated Navier-Stokes equations. The aim of this project is to establish global uniform bounds on the velocities inside the container for weak solutions to these equations with zero external force.
This involves three stages: An introduction into the theory of hydrodynamics with the aim to derive the equations of Navier-Stokes. Second, learning De Georgi’s method for regularity of solutions of elliptic equations. Finally, the research component will involve adapting these techniques to establish new regularity results and to prove the bounds on weak solutions mentioned.