When dealing with PDEs arising in fluid mechanics, bounded-energy weaksolutions are, in many cases, the only type of solutions for which one can guarantee global existence without imposing any restrictions on the size of the initial data or forcing terms. Understanding how to construct such solutions is also crucial for designing stable numerical schemes.

In this talk, we will explain the strategy for contructing weak solutions for the Navier-Stokes system for viscous compressible flows, emphasizing the difficulties encountered in the case of non-isotropic viscous stress tensors. In particular, I will present some results obtained in collaboration with Didier Bresch and Maja Szlenk.

In this talk, we present a mathematical model for tumour growth that incorporates viscoelastic effects. Starting from a basic system of PDEs, we gradually introduce the relevant biological and physical mechanisms and explain how they are integrated into the model. The resulting system features a Cahn--Hilliard type equation for the tumour cells coupled to a convection-reaction-diffusion equation for a nutrient species, and a viscoelastic subsystem for an internal velocity.
Key biological processes such as active transport, apoptosis, and proliferation are modeled via source and sink terms as well as cross-diffusion effects. The viscoelastic behaviour is described using the Oldroyd-B model, which is based on a multiplicative decomposition of the deformation gradient to account for elasticity alongside growth and relaxation effects.
We will highlight several of these effects through numerical simulations.
Moreover, we discuss the main analytical and numerical challenges. Particular focus will be given to the treatment of source and cross-diffusion terms, the elastic energy density, and the difficulties arising from the viscoelastic subsystem. The main analytical result is the global-in-time existence of weak solutions in two spatial dimensions, under the assumption of additional viscoelastic diffusion in the Oldroyd-B equation.
This work is based on joint work with Harald Garcke (University of Regensburg, Germany) and Balázs Kovács (University of Paderborn, Germany).

Systems of high-contrast resonators can be used to control and manipulate wave-matter interactions at scales that are much smaller than the operating wavelengths. The aim of this talk is to review recent studies of ordered and disordered systems of subwavelength resonators and to explain some of their topologically protected localization properties. Both reciprocal and non-reciprocal systems will be considered.
 

In the first half of the talk I will review the theory of nuclear magnetic resonance (NMR), leading to the Bloch-Torrey PDE. I will then describe the pulsed-gradient spin-echo method for measuring the Fourier transform of the voxel-averaged propagator of the Bloch-Torrey equation.  This technique permits one to compute the diffusion coefficient in a voxel. For complex biological tissue, as in the brain, the standard model represents spin-echo as a multiexponential signal, whose exponents and coefficients describe the diffusion coefficients and volume fractions of isolated tissue compartments, respectively. The question of identifying these parameters from experimental measurements leads us to investigate the degree of well-posedness of this problem that I will discuss in the second half of the talk. We show that the parameter reconstruction problem exhibits power law transition to ill-posedness, and derive the explicit formula for the exponent by reformulating the problem in terms of the integral equation that can be solved explicitly. This is a joint work with my Ph.D. student Henry J. Brown.

I will explain about how tubings of rooted trees can solve Dyson-Schwinger equations, and then summarize the two newer results in this direction, how to incorporate distinct insertion places and how when the Mellin transform is a reciprocal of a polynomial with rational roots, then one can use combinatorial techniques to obtain a system of differential equations that is perfectly suited to resurgent analysis.

Based on arXiv:2408.15883 (with Michael Borinsky and Gerald Dunne) and arXiv:2501.12350 (with Nick Olson-Harris).