TBA

TBA

The aim of this talk is to discuss a finite-volume scheme for the aggregation-diffusion family of equations with non-linear mobility
∂tρ = ∇ · (m(ρ)∇(U′(ρ) + V + W ∗ ρ)) in bounded domains with no-flux conditions. We will present basic properties of the scheme: existence, decay of a free, and comparison principle (where applicable); and a convergence-by-compactness result for the saturation case where m(0) = m(1) = 0, under general assumptions on m,U, V , and W. The results are joint works published in [1, 2]. At the end of the talk, we will discuss an extension to the Porous-Medium Equation with non-local pressure that corresponds to m(ρ) = ρm, U, V = 0 and W(x) = c|x|^−d−2s.

This project is joint work with Jose Carrillo (University of Oxford). 
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We study a very general class of first-order linear hyperbolic
systems that both become weakly hyperbolic and contain singular
lower-order coefficients at a single time t = 0. In "critical" weakly
hyperbolic settings, it is well-known that solutions lose a finite
amount of regularity at t = 0. Here, we both improve upon the analysis
in the weakly hyperbolic setting, and we extend this analysis to systems
containing critically singular coefficients, which may also exhibit
modified asymptotics and regularity loss at t = 0.

In particular, we give precise quantifications for (1) the asymptotics
of solutions as t approaches 0, (2) the scattering problem of solving
the system with asymptotic data at t = 0, and (3) the loss of regularity
due to the degeneracies at t = 0. Finally, we discuss a wide range of
applications for these results, including weakly hyperbolic wave
equations (and equations of higher order), as well as equations arising
from relativity and cosmology (e.g. at big bang singularities).

This is joint work with Bolys Sabitbek (Ghent).

Biomolecular condensates are membraneless assemblies of biomolecules (such as proteins or nucleic acids) formed through liquid-liquid phase separation. Many biomolecules are electrically charged, making condensates highly sensitive to the local electrochemical environment. In this talk, I will discuss our recent theoretical work on the dynamics of charged condensates and the role of salt concentration in their evolution toward equilibrium. Two-dimensional simulations of a thermodynamically consistent phase-field model reveal that salt can arrest coarsening by affecting the relative strength of interfacial energy, associated with the condensate surface, and electrostatic energy, arising from the formation of an electric double layer across liquid interfaces. At low salt concentrations, the electrostatic energy of the double layer becomes comparable to the interfacial energy, resulting in the emergence of multiple condensates with a fixed size. These results show that salt can act as a dynamic regulator of condensate size, with implications for both understanding biological organisation and modulating the behaviour of synthetic condensates.

The flow of viscous fluid through a soft porous medium exerts drag on the matrix and induces non-uniform deformation. This behaviour can become increasingly complicated when the medium has a complex rheology, such that deformations exhibit elastic (reversible) and plastic (irreversible) behaviour, or when the rheology has a viscous component, making the response of the medium rate dependent. This is perhaps particularly the case when compaction is repeated over many cycles, or when additional forces (e.g. gravity or an external load) act simultaneously with flow to compact the medium, as in many industrial and geophysical applications. Here, we explore the interaction of viscous effects with elastic and plastic media from a theoretical standpoint, focussing on unidirectional compaction. We initially consider how the medium responds to the reversal of flow forcing when some of its initial deformation is non-recoverable. More generally, we explore how spatial variations in stress arising from fluid flow interact with the stress history of the sample when some element of its rheology is plastic and rate-dependent, and characterise the response of the medium depending on the nature of its constitutive laws for effective stress and permeability.