Asymptotic Behaviors of systems of PDEs arising in physics and biology - 6th edition

29 Jun 2026, 08:00 → 3 Jul 2026, 18:00 Europe/Paris

Amphi 006 - Bâtiment C15 (Département de Chimie)

Aim and scope

The main goals of this workshop are the theoretical study of asymptotic behaviors (in large time or with respect to some parameters) of problems arising in physics and biology and the development of asymptotic preserving numerical methods.

The sixth edition of this workshop features nine plenary speakers. In addition, several contributed talks and a poster session will complete the program.

Plenary speakers

• Joackim Bernier (CNRS, Université de Nantes)
• José A. Cañizo (University of Granada)
• Jessica Guerand (Université de Montpellier)
• Katharina Hopf (WIAS Berlin)
• Laura Kanzler (CNRS, Paris Sorbonne Université)
• André Schlichting (University of Ulm)
• Nicolas Seguin (Inria, Université de Montpellier)
• Endre Süli (University of Oxford)
• Marie-Hélène Vignal (Université de Lyon)

Registration

Registration is now open!

Call for contributions

In addition of the plenary speakers, it is possible to apply for a contributed talk or a poster. For this, please check the corresponding buttons in the registration form and provide a .tex file of your abstract within your registration.

Important dates

• Deadline for abstracts: April 27th 
• Notification of acceptance: May 4th
• Deadline for registration: May 29th

 

Monday 29 June

13:30 Welcome

1 Navier--Stokes--Fokker--Planck Systems: Existence and Equilibration of Global Weak Solutions

Since the pioneering contributions of Werner Kuhn, Hans Kramers, Pierre-Gilles de Gennes and other scientists working at the interface of polymer chemistry and statistical physics, kinetic models have been widely and successfully used to describe the motion of polymeric fluids. During the past two decades significant progress has been made with the mathematical analysis of kinetic models of dilute polymers that involve the coupling of the incompressible or compressible Navier--Stokes equations to a Fokker--Planck equation. We shall review some of the recent developments concerning the existence of large-data global weak solutions to these models and discuss the long-time asymptotic behavior and equilibration of weak solutions.

Speaker: Endre Süli (University of Oxford)

2 Relative Entropy and Hilbert Expansion Methods in Asymptotic Limits of Kinetic–Field Systems

We investigate asymptotic regimes for kinetic equations coupled with nonlinear field effects. By combining a relative entropy framework with a modified Hilbert expansion and a careful analysis of possible boundary layers, we obtain quantitative control of remainder terms and identify the effective macroscopic dynamics in suitable scaling limits. The approach emphasizes the role of entropy dissipation and structural stability in managing multiscale interactions between transport, collisions, and nonlinear coupling mechanisms.

Speaker: Mohamed Lazhar Tayeb (Université de Tunis El Manar)

16:00 Coffee break

"Ice breaker" Poster session

Alejandro Barea Moreno (Univ. Vienna)
Hizia Bounadja (Univ. Houari Boumediene)
Abdoul Aziz Diallo (Univ. Lille)
Sebastian Tapia Mandiola (Univ. Lille)
Niccolò Tassi (Univ. Granada)
Giscard Leonel Zouakeu Wouadji (Univ. Montpellier)
Flora Philipp (TU Vienna)
Nourelhouda Khedhiri (Faculty of Science of Tunis)
Giacomo Vizzari (Univ. Toulon)
Simon Loin (Univ. Picardie Jules Verne)

3 A kinetic model of quorum sensing

Speaker: Simon Loin

abstract_quorum.pdf

4 A strong approximation in L2 for the solutions of the Maxwell system with highly oscillating periodic coefficients.

Speaker: Nourelhouda Khedhiri

RESUME.pdf

5 Analyse de modèles d’écoulements multiphasiques

Speaker: Giscard Leonel Zouakeu Wouadji

abstract-1.pdf

6 Chemotaxis Compressible Navier–Stokes Equations Modeling Vascular Network Formation

Speaker: Flora Philipp

abstract_Philipp.pdf

7 Does Speed Matter? Investigating the Role of Cell Speed Heterogeneity in Collective Migration

Speaker: Alejandro Barea Moreno

poster_alejandro_barea_moreno.pdf

8 Energy scaling for von Kármán elastic plates with positional constraints

Speaker: Giacomo Vizzari

Abstract.pdf

9 Numerical simulations of a quasilinear Gross–Pitaevskii equation with vanishing and nonvanishing conditions at inifinity

Speaker: Sebastian Tapia Mandiola

abstract.pdf

10 On decay rate of a Timoshenko System with Dual–Phase–Lag Thermoelasticity in unbounded domains

Speaker: Hizia Bounadja

abstract_Bounadja.pdf

11 Periodic solution computation for spatio-temporal conservative systems: Application to the 1D Mckean-Vlasov model

Speaker: Abdoul Aziz Diallo

abstract_ABPDE_6.pdf

12 Self-similarity and diffusive limits for linear kinetic equations: a Wild sum approach

Speaker: Niccolò Tassi

abstract-2.pdf

Tuesday 30 June

13 Self-similar behaviour of linear kinetic equations

We consider linear kinetic equations of the form $\partial_t f + \frac{1}{\epsilon} v \nabla_x f = \frac{1}{\epsilon^2} L(f)$, for an unknown $f$ which depends on time $t$, position $x$ and velocity $v$, and where $L$ is a linear operator which acts only in the velocity variable, and which typically has a probability equilibrium in $v$. Important examples include the Fokker-Planck operator, nonlocal diffusion operators, linear BGK-type operators, or linear Boltzmann operators. This PDE typically represents a mesoscopic physical model, where we keep track of the probability distribution of the position and velocity of particles. It is well known that when $\epsilon$ tends to $0$, this type of equation has a macroscopic or diffusive limit for the density $\rho(t,x) := \int f(t,x,v) dv$, which is either the standard heat equation, or the fractional heat equation. As a new result, we show that for a fixed epsilon, the behaviour of this equation for large times also follows the standard or fractional heat equation, and that the long-time and small-epsilon limits are actually interchangeable in many cases. This is a work in collaboration with Stéphane Mischler (U. Paris-Dauphine) and Niccolò Tassi (U. Granada).

Speaker: José A. Cañizo (University of Granada)

14 Kinetic and fluid equations with stiff potentials confining to a domain

In this talk, we consider two models involving extremely strong potentials, the overdamped Langevin and kinetic Fokker-Planck equations. I will present some results on the long time behavior and both on the hydrodynamic limit and on the limit when the potential becomes stiff (confining the fluid to a domain and letting boundary conditions appear). This work is motivated by the kinetic derivation of immiscible fluid equations and can be viewed as a very first step. This is an ongoing work with Frédéric Hérau (LMJL, Nantes) and Hélène Mathis (IMAG, Montpellier).

Speaker: Bérénice Grec (Université Paris Cité)

11:00 Coffee break

15 Landau damping around inhomogeneous stationary states of the Vlasov-HMF model

Plasma dynamics are often modeled by Vlasov equations, where interactions are encoded by a potential kernel. A striking phenomenon in such systems is Landau damping whereby particles relax back to a natural equilibrium state when slightly perturbed. In this talk, we investigate this phenomenon around inhomogeneous (spatially-dependent) steady states. We focus on the Vlasov-HMF (Hamiltonian Mean-Field) model, a simplification of the more central Vlasov-Poisson interaction. We first present the linearized analysis, showing how Landau damping emerges via an action-angle formulation. The core of the talk then addresses the passage from this linear picture to a nonlinear result for initial data with Sobolev regularity. This step is complicated by plasma echoes: nonlinear particle interactions that can destabilize the system. Using compact support hypotheses near the origin for symmetric perturbations, we close a bootstrap argument coupling high and low regularity norms, guaranteeing that Landau damping holds for long but finite time, up to the inverse of the perturbation size.

Speaker: Tooryanand Seetohul (Université de Rennes)

16 Time-asymptotic self-similarity of the damped Euler equations in parabolic scaling variables

In this talk, we study the long-time behavior of solutions to the compressible Euler equations with frictional damping on the whole space, assuming nonzero direction-dependent values for the density at spatial infinity. By introducing parabolic scaling variables, we reformulate the system and derive a relative entropy inequality. This framework allows us to show that the density converges to a self-similar solution of the porous medium equation, while the limiting momentum is governed by Darcy’s law. We also obtain convergence rates that explicitly depend on the flatness of the limiting profile. The main part of the analysis focuses on weak solutions in the one-dimensional case, and we further extend the results to energy-variational solutions in the multidimensional setting.
This research is joint work with Thomas Eiter (WIAS).

Speaker: Stefanie Schindler (WIAS Berlin)

17 Convergence of a finite-volume scheme for rank-deficient cross-diffusion systems via Young measure solutions

We present a Young measure approach to the convergence analysis of a fully discrete finite-volume scheme in multiple space dimensions for a class of cross-diffusion systems with a rank-deficient diffusion matrix. Such systems arise in the modelling of biological multi-component materials with incomplete diffusive mixing and may exhibit fully or partially segregated steady states.
The main analytical difficulty stems from the strongly coupled hyperbolic--parabolic structure of the system, together with a porous-medium-type degeneracy. Our analysis relies on the existence of two dissipated functionals, a Rao-type energy and the Boltzmann entropy, and on a discretisation preserving these structures. To pass to the continuum limit, we devise a Young measure framework, inspired by related developments in fluid dynamics, and introduce a vanishing artificial diffusion to identify the limiting fluxes. A weak--strong uniqueness principle then upgrades weak measure-valued convergence to strong convergence whenever the continuum model admits a strong solution.

This is joint work with Ansgar Jüngel (TU Wien).

Speaker: Katharina Hopf (WIAS Berlin)

18 Implicit-time Discontinuous Galerkin method for 1D blood flow model embedded in 3D

We present a solver for the one-dimensional blood flow equations based on a discontinuous Galerkin (DG) method in space with implicit time stepping. One-dimensional blood flow simulations are computationally efficient and useful for studying pulse wave dynamics in complex vascular networks. The implicit DG formulation ensures stability on stiff network configurations without the restrictive time-step constraints of explicit schemes, enabling efficient largescale time integration. We demonstrate the parallel performance and accuracy of the solver on benchmark problems and on realistic vascular networks.

Speaker: Raksha Devi (University of Pisa)

15:30 Coffee break

19 Quantitative Fluid Approximation for Heavy Tailed Kinetic Equations

When studying kinetic equations (which describe the motion and interaction of a system of particles), it is traditional to consider the macroscopic equations which describe the evolution of the conserved quantities of the system of particles under consideration (as for example the mass), since they are usually easier to handle. They can be obtained from the kinetic equation once an appropriate scaling and limiting procedure has been carried out. For a system of particles whose equilibrium density decays rapidly to infinity (e.g. for a Gaussian distribution), the macroscopic equation obtained is a classical diffusion equation. If the equilibrium density has a slow decay (of the algebraic type), it has been shown, so far only for equations where only one quantity is conserved, that after appropriate scaling one obtains a fractional diffusion equation.
In this talk, an extension of these results will be presented. They include the case where the linear kinetic equations considered conserve not only mass, but also momentum and energy. After appropriate scaling, a system of classical diffusion equations for the conserved quantities is obtained in the limit if the equilibrium density decays sufficiently fast to infinity. If the equilibrium density decays more slowly, we obtain fractional diffusion equations for mass and energy, while the equation for momentum is trivial. The proof is based on spectral analysis and energy estimates. They are constructive and provide explicit convergence rates.
The results of joint work with Emeric Bouin (Université Paris Dauphine) and Clément Mouhot (University of Cambridge).

Speaker: Laura Kanzler (Sorbonne Université)

20 Mean-field limit for interacting particles on adaptive networks

Collective behaviour plays a central role in many biological, physical and social systems including neuronal activity, swarming animals, opinion formation or power grids. A common modelling approach consists in describing these situations by large interacting particle systems. Moreover, many systems exhibit an underlying network structure which describes the coupling between the particles and the corresponding strength. Often, this network structure is not fixed for all time but also evolves dynamically together with the particle system.
This talk addresses such adaptively coupled interacting systems in the large particle limit. More precisely, we start with a system of finitely many particles which is coupled to an evolution equation for the underlying network. Relying on the concept of graph convergence, we derive a limiting equation for the particles' empirical measure when their number converges to infinity. This approach will lead to consider generalised particle systems with non-locality in time (memory) and non-linear network dependence.

Speaker: Sebastian Throm (Umea University)

Wednesday 1 July

21 Almost periodic solutions to nonlinear Schrödinger equations

Focusing on the case of nonlinear Schrödinger equations on the circle, I will explain why it is interesting and natural to look for almost periodic solutions for nonlinear dispersive equations on bounded domains, and why one can expect such solutions to be typical among small and smooth solutions. The point is to understand whether most small solutions of the nonlinear equation behave globally in time like those of the linear equation. I will present some recent results on the topic.

Speaker: Joackim Bernier (Université de Nantes)

22 Energy minimization for rotating Bose--Einstein condensates using a finite volume scheme

In this talk, we consider the minimization problem for the Gross–Pitaevskii energy

$E(u) = ∫_D (|∇u|² + V(x)|u|² − Ω·ū·L_z u + γ|u|⁴) dx,~~~~~~~~~~~ (E)$

under the mass constraint $‖u‖_{L²(D)} = 1$. Here, D is a domain of ℝ², V is a confining potential, $L_z = −i(x ∂_y − y ∂_x)$ is the angular momentum operator, $Ω$ is a rotational velocity parameter and $γ ≥ 0$ describes the defocusing nonlinear interactions of the condensate.
After introducing the underlying physical model and some mathematical properties in the continuous case, we will bring our attention to a finite volume discretization of $(E)$. We will begin by recalling the definition of the usual finite volume Laplacian before focusing on the angular momentum operator.
We will then state our main result, showing that minimizers of the discrete energy converge to minimizers of the continuous energy as the mesh diameter tends to zero. We then present the main ideas of the proof before showing numerical results, which notably show the presence of quantum vortices in the minimizers of $(E)$.
In the last part, we will then focus on a normalized gradient flow scheme for the computation of minimizers. In particular, we will state an exponential convergence result towards local minimizers. We will also discuss how the global phase invariance, as well as the presence or absence of rotational symmetry, affects this local convergence property and give numerical illustrations of these phenomena.
The research presented in this talk is the subject of a paper currently being written by the speaker in collaboration with Quentin Chauleur and Guillaume Dujardin.

Speaker: Robin Roussel (Université de Lille)

11:00 Coffee break

23 The Logarithmic Sobolev Inequality: Stability, Instability, and Improved Convergence Rates for the Ornstein–Uhlenbeck Flow

In certain functional inequalities, the best constants and minimizers are known. The next natural question concerns stability: if a function “almost attains equality,” in what sense is it close to a minimizer? We will discuss some recent results (both positive and negative) on quantitative stability for the Logarithmic Sobolev inequality. Our approach is based on a variant of the carré du champ method for the Ornstein–Uhlenbeck equation, enabling us to derive fully constructive estimates. This talk is based on a series of results obtained in collaboration with Giovanni Brigati (ISTA) and Jean Dolbeault (CEREMADE–Dauphine).

Speaker: Nikita Simonov (Sorbonne Université)

24 The stratified Euler equations in the sharp stratification limit

In the study of oceanic flows at the geophysical scale, the phenomenon of density stratification plays a central role in the dynamics of the system. Two categories of mathematical models are commonly used to describe the role played by the density stratification: on the one hand, continuously stratified models - such as the stratified Euler equations in a strip, considered in the present article - offer an accurate description of vertical effects, but come with a high level of complexity, both at the theoretical and numerical levels. On the other hand, bilayer models approximate the stratification by a piecewise constant profile. In the latter case, the main point is to study the evolution of the free interface between both layers, which leads to a substantially simplified model. During this talk, we will compare both approaches in the framework of the linearized stratified Euler equations around density profiles that are close to piecewise constant profiles, and prove the convergence towards the bilayer Euler equations. If time allows, we will study the effect of a shear flow, which is an intermediate step between the previous linear study and the non-linear model, through a numerical approach.

Speaker: Théo Fradin (Université de Bordeaux)

14:00 Free afternoon

Thursday 2 July

25 Gradient flow structure of the DLSS equation via diffusive transport and its structure-preserving discretization

We introduce a diffusive transport metric between probability measures that generalizes the Hellinger, Kantorovich and martingale transport.
We represent several classes of parabolic PDEs as metric gradient flows with respect to diffusive transport, such as linear second-order diffusion equations in non-divergence (Itô) form, the quadratic porous medium equation, and the fourth-order DLSS equation.
We propose a spatial discretization of the fourth-order nonlinear DLSS equation on the circle and the line, based on its diffusive gradient flow structure. The discrete dynamics inherits this gradient flow structure, and additionally satisfies contractivity in the Hellinger distance and monotonicity of several Lyapunov functionals. We prove convergence of the scheme as the mesh size vanishes, and exponential convergence in time for the equation rewritten in self-similar variables.

Based on joint works with Daniel Matthes, Eva-Maria Rott, Giuseppe Savaré.

Speaker: André Schlichting (Universtity of Ulm)

26 Derivation of the fourth order DLSS equation with nonlinear mobility via chemical reactions

We provide a derivation of the fourth-order DLSS equation based on an interpretation as a chemical reaction network. We consider on the discretized circle the rate equation for the process where pairs of particles sitting on the same site jump simultaneously to the two neighboring sites, and the reverse jump where a pair of particles sitting on a common site jump simultaneously to the site in the middle. Depending on the reaction rates, in the vanishing mesh size limit we obtain either the classical DLSS equation or a variant with nonlinear mobility of power type. We identify the limiting gradient structure to be driven by entropy with respect to a generalization of diffusive transport with nonlinear mobility via evolutionary convergence for gradient systems. Furthermore, the DLSS equation with nonlinear mobility of the power type shares qualitative similarities with the fast diffusion and porous medium equations, since we find traveling wave solutions with algebraic tails and polynomial compact support, respectively.

The talk is based on joint work with Alexander Mielke (Berlin) and André Schlichting (Ulm).

Speaker: Artur Stephan (Technische Universität Wien)

11:00 Coffee break

27 From the multi-species Boltzmann equation to an isentropic two-phase flow model

Starting from the microscopic description of gas mixtures, a rigorous kinetic foundation is essential for capturing the complex interfacial dynamics often missed by phenomenological models. In this talk, we investigate the formal hydrodynamic limit of the multispecies Boltzmann equation, specifically in a regime where intra-species collisions are the dominant physical process [1]. By analyzing the asymptotic behavior as the Knudsen numbers approach zero, we derive the isentropic two-phase flow model originally proposed by Romenski and Toro [2]. This derivation provides a justification for multivelocity and multi-pressure frameworks, allowing for the explicit computation of model coefficients. Furthermore, we demonstrate how this kinetic scaling naturally accounts for the evolution of the volume fraction, bridging the gap between kinetic theory and macroscopic multiphase flow.

This is a joint work with Gabriella Puppo (Sapienza Universita di Roma) and Thomas Rey (Université Côte d’Azur).

[1] Gabriella Puppo, Thomas Rey, and Tommaso Tenna. Formal derivation of an isentropic two-phase
flow model from the multispecies boltzmann equation. Phys. Rev. E, 113:034119, 2026.
[2] Evgeniy Romenski, Dimitris Drikakis, and Eleuterio Toro. Conservative models and numerical methods for compressible two-phase flow. J. Sci. Comput., 42(1):68–95, 2010.

Speaker: Tommaso Tenna (Sapienza Università di Roma & Université Côte d’Azur)

28 On a posteriori error analysis over star-shaped elements

In this work, we revisit the results of Babuška and Suri [1] in the setting of mesh elements that are star-shaped with respect to a ball, with the goal of deriving hp-approximation estimates featuring fully computable multiplicative constants. To this end, we establish in particular an H1 extension theorem with completely explicit stability bound. This further allows us to derive Poincaré–Friedrichs-type inequalities for L2 polynomial projections with explicit dependence on the geometric parameters and on the polynomial degree.
Building on these refined estimates, we devise a reliable residual-type a posteriori error estimator for the Hybrid High-Order (HHO) discretization of the Poisson problem which is fully explicit with respect to h, p, and the chunkiness parameter ϱ. The resulting bound is robust with respect to the polynomial degree and provides a quantitative foundation for hp-adaptive strategies on general (star-shaped) polytopal meshes.
Our results help bridge the gap between the abstract theory on general meshes (see, e.g., [2]) and practical adaptive implementations that require fully computable bounds. They may also be of independent interest for other nonconforming methods based on L2 polynomial projections, such as weak Galerkin schemes.
References
[1] I. Babuška and M. Suri. The optimal convergence rate of the p-version of the finite element method. SIAM Journal on Numerical Analysis, 1987.
[2] D. A. Di Pietro and J. Droniou. The Hybrid High-Order Method for Polytopal Meshes. Springer, Cham, 2020. Modeling, Simulation and Applications, Vol. 19

Speaker: Cyrian Marczewski (Université Polytechnique Hauts-de-France)

29 Thermodynamically coherent models for three-phase transition

We present in this talk a model for the transition between three possible phases of a same compressible fluid. To do this, we extend the usual formalism based on maximizing the specific entropy of the mixture to the three-phase case and study in particular the characterization of the triple point, which corresponds to the pressure and temperature values at which the three phases can coexist. We then deduce a (non-strictly) convex entropy at thermodynamic equilibrium. Focusing on the case of tin and its liquid, beta solid and gamma solid phases, we construct a complete equation of state and study a wide range of exact solutions to the associated Euler equations. This is a joint work with Hervé Jourdren and Corentin Stéphan.

Speaker: Nicolas Seguin (Inria, Université de Montpellier)

30 Mathematical analysis of an ocean-atmosphere coupling model

In this talk, I will present a system of partial differential equations that is used in climate models for the ocean-atmosphere interface. I will first explain some physical and mathematical assumptions that are used in the derivation of the model, and in particular the important role of turbulence parametrisation in the different boundary layers. I will then give some elements for the theoretical analysis of the corresponding problem.

Speaker: Emile Deléage (Université Grenoble Alpes)

15:30 Coffee break

31 Gehring-Type Lemma for Kinetic and Ultraparabolic Equations

Gehring’s lemma states that a function satisfying a reverse Hölder inequality on suitable subdomains enjoys improved integrability. Originally introduced by Gehring in connection with open problems in the theory of quasiconformal mappings, this result has since been adapted to the study of higher integrability properties of gradients of solutions to elliptic and parabolic equations.

In this talk, I will present results obtained through different collaborations: with Cyril Imbert and Clément Mouhot for the kinetic Fokker–Planck equation, and with Francesca Anceschi and Teresa Isernia for nonlinear ultraparabolic equations. The first key step consists in establishing a Gehring-type lemma on kinetic and ultraparabolic cylindrical domains. The second step is to derive reverse Hölder inequalities for gradients of solutions using Poincaré-type inequalities, energy estimates, and integrability properties of the solutions.

Speaker: Jessica Guerand (Université de Montpellier)

32 Kinetic modeling of congestion constraints: application to bacterial collective dynamics

In this talk, I present a work in collaboration with V. Calvez. We introduce a two species kinetic model designed to capture congestion effects through a first-order formulation coupled with hard congestion complementarity conditions. We discuss some properties of the model and introduce a numerical method for its resolution.

As an application, we consider the collective motion of the social and predatory soil bacterium Myxococcus xanthus. We focus on the celebrated rippling phenomenon, in which localized back-and-forth movements of bacteria give rise to spatio-temporal wave patterns at the macroscopic scale. We build upon our core model and include biological mechanisms. In this framework, a congestion-induced pressure acts as a nonlocal signal perceived by the bacteria. Combined with a modulation of a refractory period, this mechanism leads to the emergence of periodic patterns. The qualitative behavior of the model is illustrated and analyzed through numerical simulations.

Speaker: Tino Laidin (Université de Bretagne Occidentale)

Friday 3 July

33 Virtual Element Scheme for a Fourth-Order Poisson-Nernst-Planck-Navier-Stokes System: Mass-Conservative and Non-Conservative Schemes

The Poisson–Nernst–Planck–Navier–Stokes (PNP–NS) system describes the transport of charged species in a fluid under the influence of electric fields, coupled with fluid flow dynamics. Although the classical PNP–NS framework provides a fundamental model, it faces limitations in accurately capturing complex electrokinetic phenomena, motivating the development of enhanced formulations. In this context, a Landau–Ginzburg-type continuum theory for room-temperature ionic liquids (RTILs) is considered, leading to a modified fourth-order PNP–NS system.
A fully discrete numerical scheme based on a conforming Virtual Element Method (VEM) is presented. The spatial discretization employs an H2-conforming virtual element space for the Poisson equation, an H1-conforming space for the Nernst–Planck equations, and divergence-free conforming spaces for the Navier–Stokes system. Temporal discretization is carried out using the backward Euler method, ensuring stability and robustness.
Well-posedness of the fully discrete scheme is established, together with a priori error estimates. Numerical experiments confirm optimal convergence rates in suitable Bochner spaces and demonstrate robustness in regimes characterized by low viscosity and small permittivity parameters. Furthermore, the scheme preserves key structural and physical properties of the model, including mass, energy, and entropy, at the discrete level, for both smooth and non-smooth initial data.

Speaker: Ankur Ankur (SISSA, Trieste)

34 Qualitative Behavior of Three-Species Drift-Diffusion Equations for Memristors

The memristor is a novel semiconductor device equipped with a memory due to the change of its electrical resistance. In this way, it may mimic the behavior of a synapse in the human brain. The memristor is modeled by drift-diffusion equations which describe the transport of the electron, hole, and oxygen vacancy densities. We prove the existence of global weak solutions in the presence of recombination-generation effects. Moreover, we establish time-uniform positive upper and lower bounds away from zero for the charge carrier densities and investigate the exponential stability towards stationary solutions. This is work in progress with Ansgar Jüngel.

Speaker: Tuan Tung Nguyen (TU Wien)

35 Multiphase cross-diffusion models for tissue structures

Volume-filling cross-diffusion equations for the components of a tissue structure are formally derived from mass conservation laws and force balances for the interphase pressures and viscous drag forces in a multiphase approach. The equations include Maxwell–Stefan, tumor-growth, thin-film solar cell models as well as novel volume-filling population systems. The Boltzmann and Rao entropy structures are explored. If the drag coefficients are all equal to one, the global-in-time existence of bounded weak solutions, their long-time behavior, and the weak–strong uniqueness of solutions to a regularized system are proved using entropy methods. In the general case, the resulting diffusion matrix is positively stable, ensuring local-in-time existence of solutions. Global-in-time existence of weak solutions is proved if the drag coefficients are sufficiently close to each other. This restriction is explained by the fact that the pressure forces are of degenerate type, while the drag forces are nondegenerate in the volume fractions. Numerical simulations are presented in one space dimension to illustrate the solution behavior beyond the entropy regime.

Speaker: Sara Xhahysa (TU Wien)

11:00 Coffee break

36 Asymptotic-Preserving Schemes

In many applications, multiple physical scales, both small and large, coexist. This is the case for instance in fluid mechanics and plasma physics. Generally, the smaller scales, generically denoted by ε, have a significant impact on the cost of numerical simulations because, in the absence of specialized schemes, spatial and/or temporal discretizations must resolve the smallest of these scales. As a result, the development of efficient numerical methods for solving such multiscale problems constitutes a major computational challenge.
One way to perform numerical simulations of such models at a reasonable cost is to develop asymptotic-preserving schemes. These schemes are uniformly stable with respect to the parameter ε, making it possible to use meshes that are independent of the smallest scales present. They are said to be asymptotically stable in the limit ε → 0. Moreover, in regions where ε is very small, these schemes recover a discretization of the limiting model obtained as ε tends to zero. They are then said to be asymptotically consistent. Schemes that are both asymptotically stable and consistent are called asymptotic-preserving in the limit ε → 0, because they preserve this limit.
In this presentation, I will focus on a particular scheme that preserves the quasineutral limit for the Vlasov-Poisson equations. This work is a collaboration with Alain Blaustein (INRIA Lille), Giacomo Dimarco (University of Ferrara), and Francis Filbet (University of Toulouse). I will discuss the difficulties associated with this problem, particularly those related to its numerical simulation, and show how asymptotic-preserving schemes make it possible to overcome them.

Speaker: Marie-Hélène Vignal (Université de Lyon)