The seminar meets regularly on Wednesdays at 4pm in Eckhart 202. We also have special seminars during other days. To subscribe or unsubscribe the email list, you may either go to Camp/PDE email list or contact Chris Henderson.
Interfaces or boundaries affect the formation of crystalline phases in sometimes quite dramatic ways. Examples range from the alignment of convection roles in Benard convection perpendicular to the boundary, to the robust patterning through presomites in limb formation. Mathematically, the object of interest is a moduli space of solutions to elliptic equations in unbounded domains. This moduli space contains the relation between rate of growth and crystallographic parameters such as the width and orientation of convection rolls or presomites. I will explain the role of this moduli space and give results and conjectures on its shape in examples, starting with simple convection-diffusion and phase separation problems, and concluding with Turing patterns.
In this talk, a simple non-linear model coming from kinetic theory is considered. We will see that the global well-posedness derives from the recent Holder estimates obtained by F. Golse, C. Mouhot, A. Vasseur and the speaker for kinetic Fokker-Planck equations with rough coefficients. This is a joint work with C. Mouhot.
Consider a diffusive passive scalar advected by a two dimensional incompressible flow. If the flow is cellular (i.e.\ has a periodic Hamiltonian with no unbounded trajectories), then classical homogenization results show that the long time behaviour is an effective Brownian motion. We show that on intermediate time scales, the effective behaviour is instead a fractional kinetic process. At the PDE level this means that while the long time scaling limit is the heat equation, the intermediate time scaling limit is a time fractional heat equation. We will also describe the expected intermediate behaviour in the presence of open channels.
In this talk I will explain how to obtain the existence of the master equation for mean field games with common noise by relatively simple arguments. This equation plays a central role in the theory of optimal control problems with an infinite number of players facing a a common random perturbation. This is a joint work with A. Porretta.
I will introduce the notion of stable solutions in mean field game theory: the stable solutions are locally isolated solutions of the mean field game system. We prove that such solutions exist in potential mean field games and are local attractors for some learning procedures. This is a joint work with A. Briani.