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 fourth edition of this workshop features ten plenary speakers. In addition, several contributed talks and a poster session will complete the program.
The list of plenary speakers is now complete:
The program is now available. You can find it here.
Registrations are closed.
Important : to access the conference room, a health pass will be asked. It can be the French health pass or the European COVID Certificate scheme. More details about this pass are available here.
Limited funding for the local expenses of students and young researchers is available. If you wish to apply for such support, please register and send a CV and a publication list by email to Claire Chainais.
This talk is about connecting the 1935 Onsager Reciprocity Relations with gradient flows, and interpreting a result that Alex Mielke, Michiel Renger, and I proved in 2014 in this context. I show that our mathematical statement can be interpreted as 'showing how to generalize the Reciprocity Relations to systems with nonlinear mobility', thereby giving yet another answer to the old question 'how can we generalize Onsager's result to nonlinear systems?'. This talk is suitable for an audience who knows more about physics or chemistry than about mathematics.
I will present some techniques that can be used to “turn a numerical simulation into a theorem”. More precisely, the goal is to use a fixed point argument in the neighborhood of an approximate solution which has been obtained numerically, in order to prove the existence of a true solution nearby, and to also get guaranteed and fully computable a posteriori error estimates between this true solution and the numerical approximation. As an example, I will discuss some recent results about traveling waves in the so-called DPCM model, obtained in collaboration with C. Chainais-Hillairet and A. Zurek.
In the field of population dynamics, cross-diffusion partial differential equations have gained more impact, see for instance the SKT-model by Shigesada–Kawasaki–Teramoto. However, a rigorous derivation starting from a stochastic many-particle system was still missing in the literature.
In this talk, I will show how the approach of moderately interacting particles in the meanfield limit can be used in order to derive cross-diffusion models of SKT-type starting from a stochastic interacting many-particle system.
As a byproduct of the mean-field derivation, we also study a non-local version of the underlying PDE models. These non-local PDEs represent an intermediate level between the particle
dynamics and the final cross-diffusion partial differential equation.
This talk is based on the joint work with Li Chen, Esther Daus and Ansgar Jungel "Rigorous derivation of population cross-diffusion systems from moderately interacting particle systems", Journal of Nonlinear Science, 31(6), 1-38 (2021).
In his PhD thesis, in order to devise an experiment to measure the Avogadro number (worked out later by Perrin), Einstein derived an explicit first-order expansion for the effective viscosity of a Stokes fluid with a random suspension of small rigid particles at low density.
This formal derivation is based on two assumptions: (i) there is a scale separation between the size of particles and the observation scale, and (ii) particles do not interact with one another at first order.
In modern terms, the first assumption allows to prove a homogenization result and to rigorously define a notion of effective viscosity tensor. The second assumption allowed Einstein to give an explicit first-order expansion of this effective viscosity at low density by considering particles as isolated.
This is more subtle since the effective viscosity is a nonlinear nonlocal function of the particles. The aim of this talk is to give a rigorous justification of Einstein's first-order expansion at low density in the most general setting, and discuss higher-order corrections.
This is based on a joint work with Mitia Duerinckx (Paris-Saclay).
The homogenization of a conductive medium randomly perforated with inclusions of infinite conductivity is a well-known problem thanks to the work of Vassili Zhikov. However,
the existence of an effective model is shown under assumptions on the interparticle distance, which prevents the study of clusters and dense setting. In this talk of stochastic
homogenization, we will provide a relaxed criterion ensuring homogenization relying on ideas from network approximation. This is joint work with David Gérard-Varet from Paris University.
We consider kinetic Fokker-Planck (or Vlasov-Fokker-Planck) equations on the torus with Maxwellian or fat tail local equilibria. Results based on weak norms have recently been achieved by S. Armstrong and J.-C. Mourrat in case of Maxwellian local equilibria. Using adapted Poincare and Lions-type inequalities, we develop an explicit and constructive method for estimating the decay rate of time averages of norms of the solutions, which covers various regimes corresponding to subexponential, exponential and superexponential (including Maxwellian) local equilibria. As a consequence, we also derive hypocoercivity estimates, which are compared to similar results obtained by other techniques.
Staggered schemes have been used for a long time for the numerical simulation of incompressible and compressible flows.
They are popular because of their inherent stability and asymptotic preserving properties. However, their mathematical analysis is rather recent, and rendered more difficult because of the staggered arrangement of the unknowns.
We have recently developed some tools which generalize the famous Lax Wendroff theorem for colocated or staggered finite volume convection operators.
We apply them to a class of staggered schemes which we have been studying for some years for the numerical simulation of compressible flows, and show that any bounded converging limit of one of these schemes is a weak solution to the original problem.
Joint work with Thierry Gallouët and Jean-Claude Latché.
In this work, we consider an advection-diffusion equation, coupled to a Poisson equation for the velocity field. This type of coupling is typically encountered in applications, such as the coupling between the electric field and the electron-ion densities in plasma physics, or Darcy’s law for porous media flow. For these applications, the source term for the Poisson equation is usually zero almost everywhere in the domain, except for a small region for which a very steep source term is present. In this work, we propose a post-processing of the velocity field, which allows us to obtain a second-order numerical scheme which is robust, despite the steepness of the source term.
Few discoveries have shaped our modern society like semiconductors have. Despite of (or rather due to) several decades of research, several emerging semiconductor technologies and materials have the potential for disruptive innovation. Among them are highly-efficient perovskite solar cells, resource-efficient nanowires as well as quantum technologies. Starting from a classical nonlinear drift-diffusion model for charge transport via electrons and holes in a self-consistent electric field (the van Roosbroeck system), we present several necessary extensions. Mathematically, the challenges amount to (i) including thermodynamically consistently nonlinear diffusion in discretizations; (ii) modeling and simulating considerably slower ion movement and surface effects; (iii) coupling hyperelasticity with charge transport; (iv) incorporating atomistic effects into a macroscopic model and (v) solving an inverse problem.
A reaction-kinetic model for a two-species gas mixture undergoing pair generation and recombination reactions is considered on a flat torus. For dominant scattering with a non-moving constant-temperature background the macroscopic limit to a reaction-diffusion system is carried out. Exponential decay to equilibrium is proven for the kinetic model by hypocoercivity estimates. This seems to be the first rigorous derivation of a nonlinear reaction-diffusion system from a kinetic model as well as the first hypocoercivity result for a nonlinear kinetic problem without smallness assumptions. The analysis profits from uniform bounds of the solution in terms of the equilibrium velocity distribution.
Homogenisation theory allows to encapsulate the effective behaviour of heterogeneous materials in special averaged quantities called homogenised coefficients. in this talk, I will study the behavior of these coefficients for (random) two phases media in the dilute regime, i.e when the volume fraction of one of the phases is small.
More precisely, I will investigate a dilation model where inclusions are distributed in a constant background along a stationary ergodic point process dilated by a factor L. I will show that the associated homogenised Coefficient depends analytically on L^−1 in the dilute regime L >> 1.
The approach, that I will outline, relies on a fixed point formulation for the corrector in term of the so-called single inclusion solution and holds without the need of any quantitative theory.
In this talk, we are interested in the problem of rigorously deriving hydrodynamic equations from the Boltzmann equation for inelastic hard spheres with small inelasticity. One of the main difficulty is to identify the relation between the restitution coefficient (which quantifies the energy loss at the microscopic level) and the Knudsen number that allows us capture nontrivial hydrodynamic behavior. In this (nearly elastic) regime, we prove a result of convergence of the inelastic Boltzmann equation towards some hydrodynamic system which is an incompressible Navier-Stokes-Fourier system with self-consistent forcing terms. This is a joint work with Ricardo Alonso and Bertrand Lods.
Moment models are successfully used to simulate rarefied gases. They are hyperbolic balance laws that can be stiff with several spectral gaps, especially if the relaxation time significantly varies throughout the spatial domain. We perform a detailed spectral analysis of the semi-discrete model that reveals the spectral gaps. Based on that, we show the inefficiency of standard time integration schemes expressed by a severe restriction of the CFL number. As asymptotic preserving scheme, we use projective integration, which is an explicit scheme that is tailored to stiff multi-scale problems with large spectral gaps between one slow and (one or many) fast eigenvalue clusters. We then develop the first spatially adaptive projective integration schemes to overcome the prohibitive time step constraints of standard time integration schemes. The new schemes use different, possibly asymptotic preserving time integration methods in different parts of the computational domain, determined by the spatially varying value of the relaxation time. We use our analytical results to derive accurate stability bounds for the involved parameters and show that the severe time step constraint can be overcome. The new adaptive schemes can obtain a large speedup with respect to standard schemes.
Myxobacteria are rod-shaped, social bacteria that are able to move on flat surfaces by ’gliding’ and form a fascinating example of how simple cell-cell interaction rules can lead to emergent, collective behavior.
In this talk a new kinetic model of Boltzmann-type for such colonies of myxobacteria will be introduced and investigated. For the spatially homogeneous case an existence and uniqueness result will be shown, as well as exponential decay to an equilibrium for the Maxwellian collision operator. Furthermore, a model extension including Brownian forcing in velocity direction during the free flight phase of bacteria as well as insights in its asymptotic behavior will be presented.
The methods used for the analysis combine several tools from kinetic theory, entropy methods, hypocoercivity as well as optimal transport. The talk will be concluded with numerical simulations for the spatially homogeneous case, which are confirming the analytical results.
In this talk, we extend the derivation of the Fick-relaxation BGK model, performed in [BP S12], to a polyatomic setting. The construction of the present model is based on the introduction of relaxation coefficients and by solving an entropy minimisation problem. The distribution functions of each species are described by adding a supplementary continuous variable collecting vibrational and translational energies. Finally, by using a Chapman-Enskog equation, we recover the Fick matrix, the volume viscosity and the shear viscosity under interesting conditions.
[BPS12] S. Brull, V. Pavan et J. Schneider. Derivation of BGK models for mixtures. European Journal of Mechanics - B/Fluids, 2012, p. 74-86.
In [1] Christian Bataillon et al. have proposed a DPCM (Poisson Coupled Diffusion Model) to describe the corrosion process that occurs on the surface of steel containers in contact with the claystone formation. The model in question focuses on the development of a dense oxide layer in the region of contact between the metal and the claystone. The system formed by the layer, the metal and the solution involves the exchange and the transport of several species: electrons, iron cations and oxygen vacancies. This model leads to a system of drift-diffusion equations for the transport of charge carriers and a Poisson equation for the electrostatic potential, posed in a domain with moving boundaries. So far, some numerical methods have been proposed for the model, however no existence result has yet been established.
In our project we explore some as minor as possible corrections to make the DPCM free energy diminishing. To this aim we start by studying the model in a simplified case: we only take into account the exchanges of electrons and iron cations in a fixed oxyde layer. Our first goal is to establish a thermodynamically coherent model. To that end, some modifications on the boundary conditions are needed. Then we prove a global existence result by adapting to our needs some techniques of [2]. By cutting off all the non-linearities of the starting problem at a certain level, we obtain a regularized problem. Its solvability is proved by the investigation of systems which result from the regularized problem by a discretization of time. Finally, estimates independent of such level are established making use of the Moser iterations technique (cf. [3]). Consequently, a solution of the regularized problem will be a solution of the initial problem if the cut-off level is chosen sufficiently large.
This project has received funding from the European Union’s Horizon 2020 research and innovation programme 2014-2018 under grant agreement N°847593 (EJP EURAD).
[1] C. Bataillon, F. Bouchon, C. Chainais-Hillairet, C. Desgranges, E. Hoarau, F. Martin, S. Perrin, M. Tupin, and J. Talandier. Corrosion Modelling of Iron Based Alloy in Nuclear Waste Repository. Electrochimica Acta, 55(15):4451-4467, 2010.
[2] H. Gajewski and K. Gr¨oger. Semiconductor Equations for variables Mobilities Based on Boltzmann Statistics or Fermi-Dirac Statistics. Mathematische Nachrichten, 140(1):7-36, 1989.
[3] J. Moser. A New Proof of De Giorgi’s Theorem Concerning the Regularity Problem for Elliptic Differential Equations. Communications on Pure and Applied Mathematics, 13(3):457-468, 1960.
Biofilms are accumulations of microorganisms that can be found on almost every surface, for instance as dental plaque on teeth or as an accumulation of Staphylococcus Aureus on catheters. Based on the biofilm growth model formulated by Eberl, Parker and van Loosdrecht in 2000, we formulate an implicit Euler finite–volume scheme for the degenerate–singular diffusion equation for the biomass fraction, which is coupled with a diffusion equation for the nutrient concentration. The major challenge is the preservation of the upper bound of the biomass fraction due to the degenerate–singular diffusion equation. Our main results are the existence and uniqueness of a discrete solution and the convergence of the scheme, where the bounds of the biomass fraction and the nutrient concentration are preserved.
When designing tokamak fusion reactors, two sets of particles need to be modeled. The electromagnetically constrained plasma, which harbors the reaction, is generally modeled as a fluid. This fluid model is coupled with a kinetic equation modeling neutral particles. When the collision rate of neutrals with the background is high, a well-defined limiting equation exists. High-dimensionality of the position-velocity phase-space means that particle-based Monte Carlo becomes a go-to approach in many cases. These methods become very expensive, however, when approaching the high-collisional limit as small time steps are required to resolve the collision dynamics.
The multilevel Monte Carlo method is a method that combines simulations with coarse time steps and simulations with fine time steps in order to perform simulations with the accuracy of the fine time steps at a reduced computational cost. In this poster we show how asymptotic-preserving schemes and the multilevel Monte Carlo method can be combined to drastically reduce the computational cost of the considered simulations of neutral particle models. We will also demonstrate the achieved speed-up through numerical results.
Here we are interested in the boundary control problem of the small-amplitude water waves system in a rectangular tank. The model actually we used here is a fully linear and fully dispersive approximation of Zakharov-Craig-Sulem formulation constrained in a rectangle, in particular, with a wave maker. The wave maker acts on one lateral boundary, by imposing the acceleration of the fluid in the horizontal direction, as a scalar input signal.
Firstly, we introduce the Dirichlet to Neumann and Neumann to Neumann maps, asscociated to the certain edges of the domain, so that the system reduces to a well-posed linear control system. Then we consider the stabilizability issue on the gravity and gravity-capillary waves. It turns out that, in both cases, there exists a feedback functional, such that the corresponding control system is strongly stable. Finally, we consider the asymptotic behaviour of the above system in shallow water regime, i.e. the horizontal scale of the domain is much larger than the typical water depth. We prove that the solution of the water waves system converges to the solution of the one dimensional wave equation with Neumann boundary control, when taking the shallowness limit. Our approach is based on a detailed analysis of the Fourier series and the dimensionless version of the evolution operators mentioned above, as well as a scattering semigroup and the Trotter-Kato approximation theorem. This is a joint work with M. Tucsnak (Bordeaux) and G. Weiss (Tel Aviv).
We consider the Boltzmann equation that models a polyatomic gas by taking into account the continuous microscopic internal energy I. In particular, we consider the kinetic system proposed by [2], which is based on the procedure of Borgnakke and Larsen [1]. We linearize the Boltzmann equation around the Maxwellian function, which represents the equilibrium distribution function. Under some convenient assumptions on the collision cross-section B, we prove that the linearized Boltzmann operator L is a Fredholm operator. For this, we write L as L = K − ν.I, and we prove that K is a compact operator. The compactness is achieved as a result of K being a Hilbert-Schmidt integral operator.
This work was indeed done by Grad [3] for a monoatomic single gas, and by Pavic [4] for a mixture of monoatomic gases.
Population dynamics can be modelled by stochastic interacting any particle systems. However their numerical approximation is time consuming so one prefers to investigate simpler macroscopic models, which are derived from the many particle systems in the mean field limit. When several species are involved, this leads to nonlocal cross-diffusion terms. We investigate a nonlocal cross-diffusion system obtained in the mean field limit from such a particle system. Using the entropy method we prove existence of solutions and reveal the double entropy structure this system exhibits. Additionally we show that the nonlocal model reduces to a local model when the convolution kernels converge to a delta distribution.
In this poster, we present a Hybrid Finite Volumes scheme to discretise semiconductors models. This type of finite volumes scheme [1] is devised to handle general polygonal/polyhedral meshes, alongside with anisotropic diffusion tensors. Especially, the scheme introduced here can be used in situations where the semiconductor is immersed in a magnetic field [2].
The scheme is based on the nonlinear discretisation introduced in [3], and its analysis relies on the preservation of a discrete entropy structure, which mimics the continuous behaviour of the system. Especially, the scheme is asymptotic preserving in long-time.
Numerical experiments will highlight the main properties of the scheme.
Work in collaboration with Claire Chainais-Hillairet, Maxime Herda and Simon Lemaire.
[1] R. Eymard, T. Gallouët, and R. Herbin, Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes. SUSHI: a scheme using stabilization and hybrid interfaces, IMA J. Numer. Anal. 30 (2010), 1009–1043.
[2] H. Gajewski, and K. Gärtner, On the Discretization of van Roosbroeck’s Equations with Magnetic Field, ZAMM - Journal of Applied Mathematics and Mechanics 11 (1995), 247–264.
[3] C. Chainais-Hillairet, M. Herda, S. Lemaire, and J. Moatti, Long-time behaviour of hybrid finite volume schemes for advection-diffusion equations: linear and nonlinear approaches, Submitted for publication
Perovskite solar cells (PSCs) have become one of the fastest growing photovoltaic technologies within the last few years, for example perovskite/silicon tandem cells have become more efficient than single junction silicon solar cells [1]. However, which exact physical operation mechanisms play a fundamental role within such devices is not fully understood yet. Experiments indicate that on the one hand besides the movement of electric carriers, ion movement within the perovskite and on the other hand surface effects need to be taken into account. For this reason it is paramount to understand the electronic-ionic charge transport and the effects at inner interfaces within PSCs better via improved modelling and simulation.
In our contribution, we present a new drift-diffusion model for the charge transport in PSCs based on fundamental principles of thermodynamics and statistical physics [2]. Further, we introduce a finite volume based solver and corresponding simulations to underline the importance of our modelling approach.
1] A. Al-Ashouri et al., Monolithic perovskite/silicon tandem solar cell with >29% efficiency by enhanced hole extraction, Science 370 (6522) (2020), 1300–1309.
[2] D. Abdel, P. Vágner, J. Führmann and P. Farrell. Modelling charge transport in perovskite solar cells: Potential-based and limiting ion depletion, Electrochimica Acta 390 (2021).
Abstract in attachment
Spectral methods, thanks to their high accuracy and the possibility to use fast algorithms, represent an effective way to approximate the collisional kinetic equations of Boltzmann type, such as the Boltzmann-Nordheim equation. This equation, modeled on the seminal Boltzmann equation, describes using a statistical physics formalism the time evolution of a gas composed of bosons or fermions. Using the spectral-Galerkin algorithm introduced in [FHJ12], together with some novel parallelization techniques, we investigate some of the conjectured properties of the large time behavior of the solutions to this equation. In particular, we are able to observe numerically both Bose-Einstein condensation and Fermi-Dirac relaxation.
This work is partially funded by Labex CEMPI (ANR-11-LABX-0007-01) and ANR Project MoHyCon (ANR-17-CE40-0027-01).
I will discuss the low Mach and low Froude numbers limit for the compressible Navier-Stokes equations with degenerate, density-dependent, viscosity coefficient, in the strong stratification regime. The talk is based on a joint paper with Francesco Fanelli from Univ. Lyon. Our main result is the proof of convergence to the generalised anelastic approximation, which is used extensively to model atmospheric flows. We considered the case of a general pressure law with singular component close to vacuum, general ill-prepared initial data, and periodic boundary conditions.
In this talk, we provide a result on the derivation of the incompressible Navier-Stokes-Fourier system from the Landau equation for hard, Maxwellian and moderately soft potentials. To this end, we first investigate the Cauchy theory associated to the rescaled Landau equation for small initial data. Our approach is based on proving estimates of some adapted Sobolev norms of the solution that are uniform in the Knudsen number. These uniform estimates also allow us to obtain a result of weak convergence towards the fluid limit system.
In this talk, we will introduce the Vlasov-Navier-Stokes system, which is a fluid-kinetic model describing a cloud of particles sedimenting in a fluid. We will present some recent developments about the large time behaviour of global weak solutions to this system, when one considers the absorption of the particles at the physical boundary. We will explain how the combined effect of the absorption and the gravity leads to decay in time estimates for the solutions to the system. This result is based on [1], in the continuation of [2].
[1] L. Ertzbischoff. Decay and absorption for the Vlasov-Navier-Stokes system with gravity in a half-space. arXiv preprint arXiv :2107.02200, 2021.
[2] L. Ertzbischoff, D. Han-Kwan, and A. Moussa. Concentration versus absorption for the Vlasov-Navier-Stokes system on bounded domains. Nonlinearity, 34(10) :6843, 2021.
We consider the solution to a non-linear mean-field equation modeling a FitzHug-Nagumo neural network (see [1], [2], [3]). The non-linearity in this equation arises from the interaction between neurons. We suppose that the interactions depend on the spatial location of neurons and we focus on the behavior of the solution in the regime where short-range interactions are dominant. The solution then converges to a Dirac mass (see [4]). The aim of this talk is to characterize the blow-up profile: we will prove that it is Gaussian. Then we present interesting consequences of this result both at the macroscopic and the mesoscopic level.
[1] J. Baladron, D. Fasoli, O. Faugeras and J. Touboul. Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons The Journal of Mathematical Neuroscience, 2(10), 2012.
[2] M. Bossy, O. Faugeras and D. Talay. Clarification and complement to ”mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons”, The Journal of Mathematical Neuroscience, 5(1), 2015.
[3] S. Mischler, C. Quininao and J. Touboul. ˜ — On a kinetic FitzHugh-Nagumo model of neuronal network, Comm. Math. Phys., 342(3):1001–1042, 2015.
[4] J. Crevat, G. Faye, F. Filbet — Rigorous derivation of the nonlocal reaction-diffusion FitzHugh-Nagumo system.
The interplay between microscopic (lattice-based, SDEs, PDEs, etc.) models and macroscopic evolution equations leads to interesting questions in the classical theory of diffusion. Some of them are still in search for rigorous
answers. Upscaling of microscopic models, via a variety of techniques like renormalization, hydrodynamic limits, or homogenization, is usually a preferred methodological path. However, some useful problem settings are not the result of an averaging procedure. Hence, they require the explicit handling of models posed on separate space scales.
In this talk, we present a transport problem for signalling among plants in the context of measure-valued equations. We report on preliminary results concerning the modelling and mathematical analysis of a reaction-diffusion scenario involving the macroscopic diffusion of signalling molecules enhanced by the presence of a finite number of microscopic vesicles - pockets with own dynamics able to capture and release signals as a relay system. The coupling between the macroscopic and microscopic spatial scales relies on the use of a two-scale transmission condition and benefits of the posing of the problem in terms of measures. Mild solutions to our problem will turn to exist and will also be positive weak solutions. Preliminary numerical results will support some of our analytic results. A couple of open questions at the modeling, mathematical analysis, and simulation levels will be pointed out.
This is a joint work with Sander Hille (Leiden, NL) and Omar Richardson (Oslo, Norway) and is supported financially by the G. S. Magnussonsfond, under the auspices of the Royal Swedish Academy of Sciences.
References:
Evers, Joep; Hille, Sander; Muntean, Adrian; Measure-valued mass evolution problems with flux boundary conditions and solution-dependent velocities, SIAM Journal of Mathematical Analysis,
48 (2016), no. 3, 1929-1953.
Lind, Martin; Muntean, Adrian; Richardson, Omar; A semidiscrete Galerkin scheme for a coupled two-scale elliptic-parabolic system: well-posedness and convergence approximation rates.
BIT 60 (2020), no. 4, 999-1031.
Active fluids are suspensions of particles in viscous fluids, where particles are able to convert chemical energy into mechanical work. Compared to passive systems, active suspensions exhibit a particularly rich phenomenology. In particular, the response to an external forcing can defy intuition, with rheological measurements displaying in some settings a transition to superfluid-like behavior.
Inspired by the recent results obtained by Duerinckx and Gloria on colloidal suspensions and suspensions of sedimenting particles in a Stokes fluid, we show rigorously, by means of stochastic homogenization theory, how the presence of a random suspension of active particles (swimmers) inside a Stokes fluid can influence the viscosity. After a brief overview of the insights given by the literature of physics, I will present some results obtained in a slightly simplified setting, in which the swimming force depends linearly on the local fluid deformation.
This is joint work with Mitia Duerinckx and Antoine Gloria.
I will discuss a work in collaboration with M. Bruna (Cambdrige), M. Burger (FAU Erlangen-N¨urnberg), and S. Schulz (Wisconsin-Madison). We propose a general strategy for solving space-periodic nonlinear evolution problems with periodic boundary conditions, showing an underlying integro-differential structure, here no natural maximum/minimum principle is available. This is motivated to study several macroscopic models for active Brownian particles. To do so, we focus on a specific semilinear parabolic equation with an active drift term, which is the macroscopic model for a system of active Brownian particles with short-range and strong repulsive interactions.
Bacterial chemotaxis describes the ability of single-cell organisms to respond to chemical signals. In the case where the bacterial response to these chemical signals is sharp, the corresponding chemotaxis model for bacterial self-organization exhibits a discontinuous advection speed. This is a key challenge for analysis. We propose a new approach to circumvent the discontinuity issue following a perturbative approach, where the shape of the cellular profile is clearly separated from its global motion. As a result, we obtain exponential relaxation to equilibrium with an explicit rate. This is joint work with Vincent Calvez (Université Claude Bernard Lyon 1, France).
Hilbert’s sixth problem, stated in 1900 during the International Congress of Mathematicians, consists in the axiomatization of physics. In the case of fluid dynamics, this issue reduces to the derivation of hydrodynamic equations (a macroscopic description) from kinetic equations (a mesoscopic description), which would be themselves derived from Newton’s laws of motion applied to the particles making up the fluid (a microscopic description).
In the special case of a gas close to a global thermodynamic equilibrium with constant density, temperature and velocity, the fluctuations of these two last quantities are driven by the Navier-Stokes equations. The problem of deriving this hydrodynamic model from this kinetic model is still partially open for strong solutions (the link between weak solutions being well understood thanks to the works of C. Bardos, F. Golse, D. Levermore and L. SaintRaymond between 1989 and 2003).
Most of the strong theory of hydrodynamic limits consists in constructing solutions to the Boltzmann equation close to the solution of some hydrodynamic equation and quantifying this “closeness”. However, they require that the initial statistical distribution for the velocity decays like a Gaussian, although the ideal decay assumption, suggested by physical a priori bounds, would be an algebraic decay.
The so called Enlargement Theory (of functional spaces), initiated by C. Mouhot and developped with M.P. Gualdani and S. Mischler between 2005 and 2017, allowed to construct solutions to several kinetic equations for initial data having an algebraic decay in the velocity variable. In this talk, I will explain how this theory can be combined with previous approaches (`a la Bardos-Ukai or Gallagher-Tristani) to construct solutions to the Boltzmann equation for any initial distribution with algebraic decay, and detail the modes of convergence to the incompressible Navier-Stokes limit depending on how well prepared it is.
In this work, we study the diffusive limit approximation for a radiative heat transfer system under three different types of boundary conditions. We prove the global existence of weak solutions for this system by using a Galerkin method. Using the compactness method and Young measure theory, we prove that the weak solution converges to a nonlinear diffusion model in the diffusive limit. Under more regularity conditions on the limit system, the diffusive limit is also analyzed by using a relative entropy method. In particular, we get a rate of convergence. The initial and boundary conditions are assumed to be well-prepared in the sense that no initial and boundary layer exist. This is joint work with Xiaokai HUO (TU Wien) and Nader MASMOUDI (Courant Institute, NYUAD)
The talk is concerned with the simulation of moving rigid bodies immersed in a rarefied gas simulated by solving the Bhatnager-Gross-Krook (BGK) model for the Boltzmann equation. The computational domain for the rarefied gas changes with respect to time due to the motion of the boundaries of the rigid bodies. A one way, as well as a two-way coupling of rigid body motion and gas flow is considered.
An Arbitrary-Lagrangian-Eulerian method, where grid-points/particles are moved with the mean velocity of the gas is developed and investigated. For the spatial discretization we use a method based on a least-square approximation. For the time discretization an asymptotic preserving IMEX discretization is used. Results are compared with those of an extension of the Semi-Lagrangian numerical method suggested by Filbet & Russo to multiple space-dimensions. Moreover, the numerical results are compared with analytical, as well as with DSMC solutions of the Boltzmann equation.
Several test problems and applications illustrate the versatility of the approach.
In this talk, I will discuss the asymptotic behavior of solutions to the fast diffusion equation. It is well known that non-negative solutions behave for large times as the Barenblatt (or fundamental) solution, which has an explicit expression. In this setting, I will introduce the Global Harnack Principle (GHP), precise global pointwise upper and lower estimates of non-negative solutions in terms of the Barenblatt profile. This can be considered as the non-linear counterpart of the celebrated Gaussian estimates for the linear diffusion equations. I will characterize the maximal (hence optimal) class of initial data such that the GHP holds by means of an integral tail condition. To the best of our knowledge, analogous issues for the linear heat equation do not possess such clear answers; only partial results are known.
As a consequence, I will provide rates of convergence towards the Barenblatt profile in entropy and in stronger norms such as the uniform relative error. These estimates were fundamental in obtaining a constructive stability result in Gagliardo-Nirenberg-Sobolev inequalities. The results are based on joint work with Matteo Bonforte, Jean Dolbeault, and Bruno Nazaret.
The study of collective behaviour phenomena from a multiscale modeling perspective has seen an increased level of activity over the last years. Classical examples in socio-economy, biology, and robotics are given by self-propelled particles that interact according to a nonlinear model encoding various social rules as for example attraction, repulsion, and alignment.
Of particular interest for control design purposes is understanding the impact of control inputs in such complex systems and the study of mean-field control approaches where the control law obtain formal independence on the number of interacting agents. The construction of computational methods for mean-field optimal control is a challenging problem due to the nonlocality and nonlinearity arising from the dynamics. Furthermore, depending on the associated cost to be minimized, non-smooth and/or non-convex optimization problems might also arise.
In order to circumvent these difficulties, we propose a linearization-based approach for the computation of sub-optimal feedback laws obtained from the solution of differential matrix Riccati equations. Quantification of the dynamic performance of such control laws leads to theoretical estimates on suitable linearization points of the nonlinear dynamics. Subsequently, the feedback laws are embedded into a nonlinear model predictive control framework where the control is updated adaptively in time according to dynamic information on moments of linear mean-field dynamics. The performance and robustness of the proposed methodology are assessed through different numerical experiments in collective dynamics.
During the last decades, there has been a growing effort to understand how complex self-organised patterns (or structures) can emerge from active particle systems when the number of particles becomes very large. Typical examples in the real world include the flock of birds, the swarm of bacteria or fish schools. A few years ago, Degond et al. proposed a model of “body-orientation dynamics” where the particles carry a complex geometrical structure: the particles are modelled by rigid bodies whose attitude (or body orientation) is described by an orthonormal frame which they try to align with those of their neighbours. Using a kinetic theory approach, this talk will review and present recent results on this model regarding the many-particle limit, phase transition phenomena, pattern formation and long-time behaviour. Geometry plays an important role for all these results. This is a joint work with Pierre Degond, Amic Frouvelle, Sara Merino-Aceituno and Mingye Na.