I am a third year student on the 4-year MASDOC PhD programme at Warwick, supervised by Charlie Elliott.
My research interests lie in the field of Partial Differential Equations and its applications, and I have been concentrating in the study of evolution PDEs in non-cylindrical domains (such as moving domains or evolving hypersurfaces). This means that we look for functions that might, at distinct instants of time, be defined in different spaces. Such equations provide more realistic models for phenomena in the applied sciences, and one way to see this is by thinking of the flow of blood in the human cardiovascular system; indeed, blood is transported along arteries whose diameter and profile vary strongly due to elastic strains and to the pressure caused by the motion of the blood itself. Aside from applications, these models are also interesting (and challenging) problems from the point of view of mathematics, and new techniques are needed to treat the equations.
For my MSc dissertation at Warwick, I studied a model for the (constant mobility) Cahn-Hilliard equation on an evolving surface, for which we (Charlie and I) have established existence and uniqueness of solutions. This is a 4th order nonlinear evolution equation, and it turns out that, for the typical singular potentials (which determine the nonlinearity), global-in-time well-posedness is dependent on an interplay between the evolution of the surfaces, the Cahn-Hilliard dynamics and the initial data. As soon as possible, I will make available a pdf file containing the results we have obtained and a more extensive discussion about this problem.
At the moment, I am working towards obtaining a more unified treatment of evolution problems in these non-cylindrical domains, and extending the results one usually finds in the classical setting to the time-dependent framework. I will soon update this paragraph with my findings and more details of what I am doing!
Although I am not actively investigating either of these areas, I am also interested in the numerical analysis and modelling aspects associated with this type of equations.
At the University of Warwick (UK), as part of the MASDOC programme:
- 2019 - present: PhD candidate in Mathematics and Statistics
- Supervisor: Charlie Elliott
- 2018 - 2019: MSc in Mathematics and Statistics
- Dissertation title: Well-posedness for the Cahn-Hilliard equation on an evolving surface.
- Supervisor: Charlie Elliott
At the University of Lisbon (Portugal):
- 2016 - 2018: MSc in Mathematics
- Dissertation title: Linear stability for differential equations with infinite delay via semigroup theory.
- Supervisor: Teresa Faria
2013 - 2016: BSc in Mathematics
- Term 1, 2020/2021:
- Teaching assistant for MA259 Multivariable Calculus.
- Term 2, 2019/2020:
- Teaching assistant for MA131 Analysis II and MA4L9 Variational Analysis and Evolution Equations.
- Term 1, 2019/2020:
- Teaching assistant for MA259 Multivariable Calculus;
- Supervisor for two groups of first year students in Mathematics/Statistics.
I am a member of the SIAM-IMA Warwick Student Chapter - you can learn more about what we are doing this year by clicking here - and I am currently one of the organisers of the SPAAM seminar series. If you would like to give a talk, send me an email and we will find you a slot!
Contacts:My Warwick email is Diogo dot Caetano at warwick dot ac dot uk.