In mathematical modelling, data and solutions are often represented as measurable functions, and their quality is being captured by their membership to a certain function space. One of the core questions arising in applications of this approach is the comparison of the quality of the data and that of the solution. A particular attention is being paid to optimality of the results obtained. A delicate choice of scales of suitable function spaces is required in order to balance the expressivity (the ability to capture fine mathematical properties of the model) and the accessibility (the level of its technical difficulty) for a practical use. We will give an overview of the research area which grew out of these questions and survey recent results obtained in this direction as well as challenging open questions. We will describe a development of a powerful method based on the so-called reduction principles and demonstrate its use on specific problems including the continuity of Sobolev embeddings or boundedness of pivotal integral operators such as the Hardy - Littlewood maximal operator and the Laplace transform.

***CANCELLED*** In this talk, I will discuss recent results related to the mathematical justification of PDEs which model multi-phase flows at the macroscopic level from mesoscopic descriptions with jump conditions at interfaces. We will also present interesting and difficult open problems.

We use entropy methods to show that the heat equation with Dirichlet boundary conditions on the complement of a compact set in R^d shows a self-similar behaviour much like the usual heat equation on R^d, once we account for the loss of mass due to the boundary. Giving good lower bounds for the fundamental solution on these sets is surprisingly a relatively recent result, and we find some improvements using some advances in logarithmic Sobolev inequalities. In particular, we are able to give optimal asymptotic bounds for large times for the fundamental solution with an explicit approach rate in dimensions larger than 2, and some new bounds in dimension 2.

This is a work in collaboration with Alejandro Gárriz and Fernando Quirós.

This talk will be devoted to our recent developments in the analysis of emerging models for complex flows. I will start from presenting a general PDE system describing two-fluid flows, for which we prove existence of global in time weak solutions for arbitrary large initial data. I will explain where the famous approach of Lions developed for the compressible Navier-Stokes equations fails and how to use a more direct, weighted Kolmogorov criterion to prove compactness of approximating sequences of solutions. Through a formal limit, I will link the two-fluid model to the constrained two-phase models. Applications of such models include modelling of granular flows, crowd motion, or shallow water flow through a channel. The last part of my talk will focus on the rigorous derivation of these models from the compressible Navier-Stokes equations via the vanishing singular pressure or viscosity limit.

I will present a model for an optimal portfolio allocation and consumption problem for a portfolio composed of a risk-free bond and two illiquid assets. Two forms of illiquidity are presented, both illiquidities based on Lévy processes. The goal of the investor is to maximise a certain utility function, and the optimal utility is found as a solution of a nonlinear PIDE of the Hamilton-Jacobi-Bellman kind.