The main goal of this project is to develop mathematical methods for the mathematically rigorous analysis of the properties of interfaces evolving in a heterogeneous, random environment, described on a small scale by nonlinear PDEs with random coefficients.
It is motivated by the following situation: With an interface is associated a scalar quantity called its energy (think eg of its area) which it tries to decrease, ie, it performs a gradient flow.
This energy is perturbed through obstacles or impurities on a very small scale, and the system is driven by some large-scale force. The impurities are random, ie we have information only on the probability of finding certain impurities in a certain place, not on their precise nature and location.
We are interested in the effective velocity and other qualitative properties of the interface on a large scale, much larger than the scale on which the perturbations vary.
On that scale, the perturbations should ’average out’, but we can ask the following natural questions: What is the effective evolution law on a large scale, and what are the qualitative properties of the interface, eg on which scales does it look rough due to all the random heterogeneities? (The latter is related to the question of error estimates.) How does all this depend on the law of the impurities? You will address, depending on your interest and background, some aspects of but not necessarily all these questions.
Please contact the supervisor when you want to pursue this project, citing the project title in your email
Funding Notes
We are interested in pursuing this project and welcome applications if you are self-funded or have funding from other sources, including government sponsorships or your employer.
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