ON DEMAND PDE CDT Students & Alumni Reunion Event - DAY ONE
Dr Andreas Søjmark, London School of Economics
Cohort 2 Alumni
On temperature discontinuities in the supercooled Stefan problem with noise
In this talk, I will present a version of the supercooled Stefan problem with Brownian noise. The main part of the talk will focus on the physical derivation of this problem and how one can naturally consider the emergence of discontinuities in the temperature profile as a function of time. Moreover, I will introduce a probabilistic representation of the problem, which is used to establish an interesting probabilistic dichotomy: it turns out that there is a simple condition on the initial heat profile in relation to a single parameter for how the liquid solidifies, which, if violated, leads to temperature discontinuities in finite time with positive probability, whereas the temperature remains globally continuous in time with probability one when the aforementioned condition is satisfied.
Bogdan Raiță - Scuola Normale Superiore di Pisa
Cohort 1 Alumni
Old and new in compensated compactness
We will review aspects of the theory of Compensated Compactness, starting with the fundamental work of Murat and Tartar and concluding with recent results obtained jointly with A. Guerra, J. Kristensen, and M. Schrecker. Broadly speaking, the object of this study is to gain a better understanding of the interaction between weakly convergent sequences and nonlinear functionals. The general framework will be that of variational integrals defined on spaces of vector fields satisfying linear pde constraints that satisfy Murat's constant rank condition. We will focus on the weak (lower semi-)continuity of these integrals, as well as the Hardy space regularity of the integrands. We also describe the oscillation and concentration effects of linear pde constrained sequences using generalized Young measures.
Further research
https://link.springer.com/content/pdf/10.1007/s00205-022-01775-3.pdf
https://link.springer.com/content/pdf/10.1007/s00526-019-1544-x.pdf
Dr Matthew Schrecker, University College London
Cohort 1 Alumni
Gravitational Collapse of Self-Similar Stars
The Euler-Poisson equations give the classical model of a self-gravitating star under Newtonian gravity. It is widely expected that, in certain regimes, initially smooth initial data may give rise to blow-up solutions, corresponding to the collapse of a star under its own gravity. In this talk, I will present recent work with Yan Guo, MahirHadzic and Juhi Jang that demonstrates the existence of smooth, radially symmetric, self-similar blow-up solutions for this problem. At the heart of the analysis is the presence of a sonic point, a singularity in the self-similar model that poses serious analytical challenges in the search for a smooth solution.