EMRM 2023 – Abstracts (Invited Talks)

Miroslav Bulíček

Coupling Navier–Stokes–Fourier with the Johnson–Segalman stress-diffusive viscoelastic model: global-in-time large-data analysis

We focus on a large-data and global-in-time theory for weak solutions to a system of partial differential equations describing an unsteady flow of an incompressible heat-conducting rate-type viscoelastic stress-diffusive fluid filling up a mechanically and thermally isolated container of any dimension. To overcome the principle difficulties connected with ill-posedness of the diffusive Oldroyd-B model in three dimensions, we assume that the fluid admits a strengthened dissipation mechanism, at least for excessive elastic deformations. All the relevant material coefficients are allowed to depend continuously on the temperature, whose evolution is captured by a thermodynamically consistent equation. We present a huge zoo of models for which, just based on the energy and the entropy inequality, one can deduce well-sounded existence theory for a weak solution.

Ewelina Zatorska

Two-phase flows: challenges and insights

I will present two-phase flows, explaining their differences from mixtures and addressing the complexities they bring. I will then focus on analysis of the two-phase flow model under the algebraic pressure closure.
Throughout the talk, I will summarize key findings, including existence of global weak solutions, their regularity, uniqueness (or lack thereof), and conditional weak-strong uniqueness. I will also discuss the challenges we face in improving this conditional uniqueness result.

Julian Fischer

A rigorous approach to the Dean–Kawasaki equation of fluctuating hydrodynamics

Fluctuating hydrodynamics provides a framework for approximating density fluctuations in interacting particle systems by suitable SPDEs. The Dean–Kawasaki equation – a strongly singular SPDE – is perhaps the most basic equation of fluctuating hydrodynamics; it has been proposed in the physics literature to describe the fluctuations of the density of N diffusing weakly interacting particles in the regime of large particle numbers N. The strongly singular nature of the Dean–Kawasaki equation presents a substantial challenge for both its analysis and its rigorous mathematical justification: Besides being non-renormalizable by approaches like regularity structures, it has recently been shown to not even admit nontrivial martingale solutions.
In this talk, we give an overview of recent quantitative results for the justification of fluctuating hydrodynamics models. In particular, we give an interpretation of the Dean–Kawasaki equation as a "recipe" for accurate and efficient numerical simulations of the density fluctuations for weakly interacting diffusing particles, allowing for an error that is of arbitarily high order in the inverse particle number.
Based on joint works with Federico Cornalba, Jonas Ingmanns, and Claudia Raithel.

Virginie Ehrlacher

Boundary stabilization of cross-diffusion systems in moving domains

This work is motivated by a collaboration with the French Photovoltaic Institute. The aim of the project is to propose a model in order to simulate and optimally control the fabrication process of thin film solar cells. The production of the thin film inside of which occur the photovoltaic phenomena accounting for the efficiency of the whole solar cell is done via a Physical Vapor Deposition (PVD) process. More precisely, a substrate wafer is introduced in a hot chamber where the different chemical species composing the film are injected under a gaseous form. Molecules deposit on the substrate surface, so that a thin film layer grows by epitaxy. In addition, the different components diffuse inside the bulk of the film, so that the local volumic fractions of each chemical species evolve through time. The efficiency of the final solar cell crucially depends on the final chemical composition of the film, which is freezed once the wafer is taken out of the chamber. A major challenge consists in optimizing the fluxes of the different atoms injected inside the chamber during the process for the final local volumic fractions in the layer to be as close as possible to target profiles. Two different phenomena have to be taken into account in order to correctly model the evolution of the composition of the thin film: 1) the cross-diffusion phenomena between the various components occuring inside the bulk; 2) the evolution of the surface. As a consequence, the underlying model reads as a cross-diffusion system defined on a moving boundary domain. The complete optimal control problem of the fluxes injected in the hot chamber is currently out-of-reach in terms of mathematical analysis. The aim of this talk is to theoretically investigate a simpler problem, which is the boundary stabilization of the model used to simulate the PVD process. We show first exponential stabilization and then finite-time stabilization in arbitrary small time of the linearized system around uniform equilibria, provided the underlying cross-diffusion system has an entropic structure with a symmetric mobility matrix. This stabilization is achieved with respect to both the volumic fractions of the different chemical species composing the thin film and the thickness of the latter. The feedback control is derived using the backstepping technique, adapted to the context of a time-dependent domain. In particular, the norm of the backward backstepping transform is carefully estimated with respect to time.

Cleopatra Christoforou

A hydrodynamic model of flocking type: BV weak solutions and long-time behaviour

In this talk, we present some results on the existence and time-asymptotic flocking of weak solutions to a hydrodynamic model of flocking type with an all-to-all interaction kernel in a one-space dimension. An appropriate notion of entropy weak solutions with bounded support is given to capture the behavior of solutions with initial data that has finite total mass confined in a bounded interval and initial density uniformly positive therein. We show global-in-time existence of entropy weak solutions with concentration for any initial data of bounded variation having the structure above. In addition, we will describe a recent result on the time-asymptotic limit for such solutions, showing the asymptotic decay towards flocking profiles without any further restrictions on the data. Joint work with Debora Amadori.

Vincent Giovangigli

Asymptotic stability for a multicomponent reactive flow diffuse interface model

We investigate the asymptotic stability of constant equilibrium states for a multicomponent reactive flow diffuse interface model of Korteweg type. The model is obtained with simplifying assumptions from general Cahn–Hilliard fluid equations derived from the kinetic theory of dense gas mixtures and the BBGKY hierarchy. An augmented formulation is introduced with an extra unknown representing the gradient of density. The stability properties of the linearized augmented system as well as the validity of the gradient constraint are key arguments in the proof of asymptotic stability.

Benoît Perthame

Analysis of models of living tissues and free boundary problems

Tissue growth, as it occurs during solid tumors, can be described at a number of different scales from the cell to the organ. For a large number of cells, "fluid mechanical" approaches have been advocated in mathematics, mechanics, or biophysics. We will give an overview of the modeling aspects and focus on the links between those mathematical models. Then, we will focus on the "compressible" description describing the cell population density based on systems of porous medium type equations with reaction terms. A more macroscopic "incompressible" description is based on a free boundary problem close to the classical Hele–Shaw equation. In the stiff pressure limit, one can derive a weak formulation of the corresponding Hele–Shaw free boundary problem and one can make the connection with its geometric form. The mathematical tools related to these questions include multi-scale analysis, Aronson–Benilan estimate, uniform L⁴ estimate on the pressure gradient, and emergence of instabilities.

Klemens Fellner

Oscillatory solutions to a nonlinear Becker–Döring type model for prion dynamics

Prions are able to self-propagate biological information through the transfer of structural information from a misfolded/infectious protein in a prion state to a protein in a non-prion state. Prions cause diseases like Creuzfeldt–Jakob. Prion-like mechanisms are associated to Alzheimer, Parkinson, and Huntington diseases. We present a fundamental bi-monomeric, nonlinear Becker–Döring type model, which aims to explain experiments in the lab of Human Rezaei showing sustained oscillatory behaviour over multiple hours. Besides two types of monomers, our model suggests a nonlinear depolymerisation process as crucial for the oscillatory behaviour. Since then, experimental evidence seems to confirm this process. We provide details on the mechanism of oscillatory behaviour and show numerical simulations.

Agnieszka Świerczewska-Gwiazda

Asymptotic analysis: from high friction gas dynamics to diffusion models

Several recent studies considered the high-friction limit for systems arising in fluid mechanics. Following this approach, we rigorously derive the nonlocal Cahn–Hilliard equation as a limit of the nonlocal Euler–Korteweg equation using the relative entropy method. Applying the recent result about relations between non-local and local Cahn–Hilliard, we also derive rigorously the large-friction nonlocal-to-local limit. The proof is formulated for dissipative measure-valued solutions of the nonlocal Euler–Korteweg equation, which are known to exist on arbitrary intervals of time. This approach provides a new method to derive equations not enjoying classical solutions via relative entropy method by introducing the nonlocal effect in the fluid equation. During the talk I will also discuss the high-friction limit of the Euler–Korteweg system.

Robert Lasarzik

Energy-variational solutions for different viscoelastic fluid models

The concept of energy-variational solutions is introduced for a general class of evolution equations. Under certain convexity assumptions, the existence of such solutions can be shown constructively by an adapted minimizing movement scheme. Weak-strong uniqueness follows by a suitable relative energy inequality. Moreover, energy-variational solutions are compared to other generalized solution concepts for specific systems in fluid dynamics. Finally, the general result is applied to two different viscoelastic fluid models without stress diffusion, and a short comparison of different viscoelastic models hints at advantages and disadvantages of this energy-variational approach.