I am an applied mathematician interested in applied/numerical analysis of partial differential equations (PDEs). Recently I have been interested in the analysis and numerical analysis of fully nonlinear PDEs, such as systems arising from the theory of mean field games:
as well as free boundary problems, for example the system governing the motion of a rigid body immersed in a viscous fluid:
Prior to this my PhD research focused on PDEs posed on evolving surfaces — in particular the Cahn-Hilliard equation (and variants thereof):
C. M. Elliott and T. Sales, "A constructive approach to a fluid-rigid body interaction problem", 2026. (To appear in Interfaces and Free Boundaries) (arXiv).
C. M. Elliott and T. Sales, "An evolving surface finite element method for the Cahn-Hilliard equation with a logarithmic potential", 2026. (IMA Journal of Numerical Analysis) (arXiv)
C. M. Elliott and T. Sales, "The evolving surface Cahn-Hilliard equation with a degenerate mobility", 2026. (Nonlinear Analysis: Real World Applications) (arXiv)
C. M. Elliott and T. Sales, "A fully discrete evolving surface finite element method for the Cahn-Hilliard equation with a regular potential", 2025. (Numerische Mathematik) (arXiv)
C. M. Elliott and T. Sales, "Navier-Stokes-Cahn-Hilliard equations on an evolving surface", 2025. (Interfaces and Free Boundaries) (arXiv)
Code from the numerical experiments in these works can be made available upon reasonable request.
T. Sales, "A (tangential) Navier-Stokes-Cahn-Hilliard system on an evolving surface", 2024. In Oberwolfach Report 21, (3) "Interfaces, Free Boundaries, and Geometric Partial Differential Equations".