Formation of singularities for the relativistic Euler equations (N. Athanasiou): An archetypal phenomenon in the study of hyperbolic systems of conservation laws is the development of singularities (in particular shocks) in finite time, no matter how smooth or small the initial data are. A series of works by Lax, John et al confirmed that for some important systems, when the initial data is a smooth small perturbation of a constant state, singularity formation in finite time is equivalent to the existence of compression in the initial data. Our talk will address the question of whether this dichotomy persists for large data problems, at least for the system of the Relativistic Euler equations in (1+1) dimensions. We shall also give some interesting studies in (3+1) dimensions. This is joint work with Dr. Shengguo Zhu.
Global Well-Posedness for a Class of Stochastic McKean-Vlasov Equations in One Dimension (A. Mayorcas): We show global well-posedness for a family of parabolic McKean--Vlasov SPDEs with additive space-time white noise. The family of interactions we consider are those given by convolution with kernels that are at least integrable. We show that global well-posedness holds in both the repulsive/defocussing and attractive/focussing cases. Our strategy relies on both pathwise and probabilistic techniques which leverage the Gaussian structure of the noise and well known properties of the deterministic PDEs.
- PDE CDT Lunchtime Seminar