The Stokes–Onsager–Stefan–Maxwell (SOSM) equations model the flow of concentrated mixtures of distinct chemical species in a common thermodynamic phase. We derive a novel variational formulation of these nonlinear equations in which the species mass fluxes are treated as unknowns. This new formulation leads to a large class of high-order finite element schemes with desirable linear-algebraic properties. The schemes are provably convergent when applied to a linearization of the SOSM problem.

We propose a nematic model for polyatomic gas, intending to study anisotropic phenomena. Such phenomena stem from the orientational degree of freedom associated with the rod-like molecules composing the gas. We adopt as a primer the Curitss-Boltzmann equation. The main difference with respect to Curtiss theory of hard convex body fluids is the fact that the model here presented takes into account the emergence of a nematic ordering. We will also derive from a kinetic point of view an energy functional similar to the Oseen-Frank energy. The application of the Noll-Coleman procedure to derive new expressions for the stress tensor and the couple-stress tensor will lead to a model capable of taking into account anisotropic effects caused by the emergence of a nematic ordering. In the near future, we hope to adopt finite-element discretisations together with multi-scale methods to simulate the integro-differential equation arising from our model.

This talk presents a general framework for modelling the dynamics of limit order books, built on the combination of two modelling ingredients: the order flow, modelled as a general spatial point process, and the market clearing, modelled via a deterministic ‘mass transport’ operator acting on distributions of buy and sell orders. At the mathematical level, this corresponds to a natural decomposition of the infinitesimal generator describing the  order book evolution into two operators: the generator of the order flow and the clearing operator. Our model provides a flexible framework for modelling and simulating order book dynamics and studying various scaling limits of discrete order book models. We show that our framework includes previous models as special cases and yields insights into the interplay between order flow and price dynamics. This talk is based on joint work with Rama Cont and Pierre Degond.