This workshop aims to bring together researchers working at the intersection of mathematical biology, fluid mechanics, and the analysis of partial differential equations. The focus will be on mathematical models and analytical techniques that arise in both fields, highlighting common structures, challenges, and methods. Topics of particular interest include aggregation-diffusion equations, cross-diffusion systems, Euler equations with nonlocal interactions, and techniques for establishing singular limits in nonlinear PDEs. The workshop will provide an opportunity to exchange ideas, explore connections between different approaches, and stimulate new collaborations.

Registration 

There is no any fee however we kindly ask you to register (sooner better - that will allow us to estimate refreshments for a coffee break)

REGISTRATION 

All lectures will take place on Thursday 12th of November 2026 in room 403 on the 4th floor of IMPAN, ul. Śniadeckich 8, Warszawa.

Chair of the session is Jakub Skrzeczkowski

Schedule:

10:30 - 11:10 Charles Elbar

11:15 - 11:55 Guy Parker

12:00 - 14:30 Lunch break

14:30 - 15:10 Yurij Salmaniw 

15:15 - 15:55 Jakub Woźnicki

16:00 - 16:30 Coffee break 

16:30 - 17:10 Maja Szlenk

17:15 - 17:55 Qingyou He

Abstracts:

Charles Elbar (CMAP/Ecole Polytechnique)

Title: Global existence for a cross diffusion system with different mobilities

Abstract: We consider a cross diffusion system of two populations, often called the Busenberg-Travis system. The two species are transported by the same pressure gradient with Darcy’s law, modeling overcrowding effect (populations tend to move away from regions of high pressure). However, their mobility is different: the first species moves with mobility 1, whereas the second moves with mobility \nu. The difficulty to prove existence is to prove strong compactness of each densities, which we achieve with a variant of the div-curl lemma and Young measures.

Guy Parker (Université Claude Bernard, France) 

Title: Interfaces and non-uniqueness in a cross-diffusion system with independent drifts

Abstract: When two densities diffuse along a common pressure which depends only on their sum, species segregation persists across time and sharp interfaces are transported along the flow. However, if each density is also transported by an independent potential drift, this property can not be guaranteed. Here, I will present recent work showing that the persistence or breakdown of an inter-species interface instead depends on the chosen notion of solution. When the drifts push the two phases towards each other, the vanishing-viscosity solution creates an overlap, whereas a segregated weak solution also exists. Our analysis studies a new relative entropy variable which can measure the mixing of the two species. The talk will be based on joint work with Charles Elbar.

Yurij Salmaniw (University of Sheffield) 

Title: Qualitative and Quantitative Properties of Aggregation-Diffusion Equations on the One-Dimensional Torus

Abstract: We consider the qualitative and quantitative behaviour of aggregation–diffusion equations on the one-dimensional torus, with the goal of understanding the emergence and stability of nonhomogeneous stationary solutions. We focus primarily on the scalar equation, examining its long-time behaviour, convergence to the homogeneous state, and the role of the interaction kernel in determining stability and instability. We then introduce a fixed-point formulation of the stationary problem and use local bifurcation theory to characterize the emergence, criticality, and stability of nonhomogeneous solution branches. Numerical bifurcation diagrams illustrate these results and provide further insight into the global structure of stationary solutions. Finally, we discuss extensions to interacting populations, highlighting similarities and differences with the scalar case.

Jakub Woznicki (University of Vienna) 

Title: Existence and weak-strong uniqueness of measure solutions to Euler-alignment/Aw--Rascle--Zhang models

Abstract: We study the multi-dimensional Euler-alignment system with a matrix-valued communication kernel, motivated by models of anticipation dynamics in collective behaviour. A key feature of this system is its formal equivalence to a nonlocal variant of the Aw--Rascle--Zhang (ARZ) traffic model, in which the desired velocity is modified by a nonlocal gradient interaction. We prove the global-in-time existence of measure solutions to both formulations, obtained via a single degenerate pressureless Navier--Stokes approximation. Furthermore, we establish a weak-strong uniqueness principle adapted to the pressureless setting and to nonlocal alignment forces. As a consequence, we rigorously justify the formal correspondence between the nonlocal ARZ and Euler-alignment models: they arise from the same inviscid limit, and the weak-strong uniqueness property ensures that, whenever a classical solution exists, both formulations coincide with it.

Qingyou He (Laboratoire Jacques-Louis Lions, Sorbonne Université) 

Title:  Some sharp geometric properties of two related reaction-diffusion type free boundary problems 

Abstract: The porous medium equation and the Hele-Shaw problem are two classical diffusive mechanisms related to the free boundary problems, which are connected by the incompressible limit. Taking the additional reaction terms into consideration, this is the so-called the reaction-diffusion equation, where the free boundary is referred to the support boundary. In this talk, I will present the sharp Lipschitz continuity for porous medium type reaction-diffusion equation and sharp power concavities for the two free boundary problems. The interest is that we can see some geometric differences for the two related free boundary problems.

Maja Szlenk (IMPAN)

Title: Discontinuous shear-thickening asymptotic for power-law compressible flows

Abstract: I will present my recent work with Didier Bresch and Cosmin Burtea. We study the convergence of a power-law model for dilatant compressible fluids to a class of models exhibiting a maximum admissible shear rate, called thick fluids, which are used for example to model solid particles suspended in a fluid with a high particle-to-fluid volume fraction, and the phenomenon of jamming. Mathematically, these types of materials are model by taking the limit of a power-law system, when the power-law exponent goes to infinity. We characterize this limit in two particular cases - for the Stokes system in multiple dimensions, and for the full Navier-Stokes system in one dimension.