In this talk we consider two large groups of interacting agents, whose dynamics are influenced by the overall perceived density. Such dynamics can be used to describe two pedestrian groups, walking in opposite directions. Or to model the relocation behavior of two distinct populations, which have a preference to stay within their own group.
We discuss the mathematical modeling in different...
An implicit Euler finite-volume scheme for a degenerate cross-diffusion system describing the ion transport through biological membranes is proposed. We consider the model developed in [1] for describing size exclusion effects in narrow channels. The strongly coupled equations for the ion concentrations include drift terms involving the electric potential, which is coupled to the...
In this talk we study a PDE model for two diffusing species interacting by local size exclusion and global attraction. This leads to a nonlinear degenerate cross-diffusion system, for which we provide a global existence result as well as a uniqueness proof in the case of equal diffusivities. The analysis is motivated by the formulation of this system as a formal gradient flow for an...
Existence of global in time radially symmetric solutions is studied for "large" initial data.
Criteria for blowup of solutions in terms of Morrey norms are derived.
Inspired by so-called TVD limiter-based second-order schemes for hyperbolic conservation laws, we develop a second-order accurate numerical method for multi-dimensional aggregation equations. The method allows for simulations to be continued after the first blow-up time of the solution. In the case of symmetric, lambda-convex potentials with a possible Lipschitz singularity at the origin we...
Conservation laws in continuum physics are often coupled, for example the continuity equations for a reacting gas mixture or a plasma are coupled through multi-species diffusion and a complicated reaction mechanism. For space discretisation of these equations we employ the finite volume method. The purpose of this talk is to present novel flux vector approximation schemes that incorporate this...
The run-and-tumble motion of bacteria such as E. coli can be represented by a nonlinear kinetic equation. It will be considered under an hyperbolic scaling, and rewritten using the Hopf-Cole transform of the distribution function. It has been shown that the asymptotic model is either a Hamilton-Jacobi equation in which the Hamiltonian is implicitely defined, or a non-local Hamilton-Jacobi-like...
We will discuss a collective behavior model in which individuals try to imitate each other’s velocity and have a preferred asymptotic speed. It is a variant of the well-known Cucker-Smale model in which the alignment term is localized. We showed that a phase change phenomenon takes place as diffusion decreases, bringing the system from a disordered'' to anordered'' state. This effect is...
In this talk, we will present a recent result on fluid solid interaction problem. We consider the system formed by the incompressible Navier Stokes equations coupled with Newton’s laws to describe the motion of a finite number of homogeneous rigid disks within a viscous homogeneous incompressible fluid in the whole space R2 . The motion of the rigid bodies inside the fluid makes the fluid...
A widely used prototype phase model to describe the synchronous behavior of weakly coupled limit-cycle oscillators is the Kuramoto model whose dynamics for sufficiently large ensemble of oscillators can be effectively approximated by the corresponding mean-field equation ’the Kuramoto Sakaguchi Equation’. In the recent past, it has been extensively studied to analyze the phase transition of...
The classical Nernst-Planck model suffers from its inability to accurately resolve boundary layers where locally large ion concentrations and pronounced voltage differences occur. In nanofluidic applications like nanopores with charged pore walls,
one spatial dimension is in the order of the Debye length which corresponds to the boundary layer width.
Improved models, in particular models that...
To model hybrid resonances in fusion plasma, Maxwell’s equations feature a sign changing permittivity tensor. The problem can be expressed as a degenerate elliptic PDE. There is no uniqueness of the solution, and the solutions admit a singularity inside the domain.
A small regularizing viscosity parameter can be introduced, but the problem is still numerically challenging because of the...
We consider the sedimentation of N identical spherical particles in a uniform gravitational field. Particle rotation is included in the model while inertia is neglected.
In the dilute case, the result in [5] shows that the particles do not get closer in finite time. The rigorous convergence of the dynamics to the solution of a Vlasov-Stokes equation is proven in [4] in a certain averaged...
The Prandtl equation was derived in 1904 by Ludwig Prandtl in order to describe the behavior of fluids with small viscosity around a solid obstacle. Over the past decades, several results of ill-posedness in Sobolev spaces have been proved for this equation. As a consequence, it is natural to look for more sophisticated boundary layer models, that describe the coupling with the outer Euler...
The aim of this work is the study of out-of-equilibirum plasma physics. It is a multiscale problem involving both very small lengths (Debye length) and high frequency oscillations (electronic plasma frequency). Transport of charged particles (electrons and ions) in context of Inertial Confinement Fusion (ICF) can be modelled by the bi-temperature Euler equations, which are a non-conservative...
Interface processes play an important role in many electrochemical applications like batteries, fuel-cells or water purification. In boundary regions typically sharp layers form where electrostatic potential develops steep gradients and the ionic species accumulate to an extend that saturation effects become relevant. In contrast, the classical Nernst-Planck model for electrolyte transport is...
Numerous slurry transportation pipeline systems have been built in the past 10 years. At the same time, T. Chakkour & F. Benkhaldoun study in [2, 3] the hydraulic transport of particles in tubes. We investigated in [1] the hydraulic transport of slurry system in horizontal tubes (The Khouribga mines). This mineral pipeline has often been referred to as one of the most challenging projects in...
We consider two different partial differential equation models structured by elapsed time for dynamics of neuron population and give some improved results for long time asymptotics. The first model we study is a nonlinear version of the renewal equation, while the second model is a conservative drift-fragmentation equation which adds adaptation and fatigue effects to the neural network model....
It is the purpose of this talk to provide an overview on recent advances on the development of Lagrange Projection like numerical schemes for compressible fluids systems with source terms.
The key idea of the Lagrange-Projection strategy is to decouple the acoustic and transport phenomenon. When combined with a Suliciu like relaxation technique, the Lagrange-Projection strategy leads to...
We discuss ideas and tools to construct Lyapunov functionals on the space of probability measures to investigate convergence to global equilibrium of partially damped Euler equations under the influence of external and interaction potential forces with respect to the 2-Wasserstein distance.
This talk aims at illustrating some methods for the asymptotical analysis of optimal contribution-ol problems. We use examples in the context of groundwater pollution. The spatio-temporal objective takes into account the economic trade off between the pollutant use –for instance fertilizer– and the cleaning costs. It is constrained by a hydrogeological PDEs model for the spread of the...
We study the qualitative convergence behavior of a novel FV-discretization scheme of the Fokker-Planck equation, the squareroot approximation scheme (SQRA), that recently was proposed by [Lie, Fackeldey and Weber 2013] in the context of conformation dynamics. We show that SQRA has a natural gradient structure related to the Wasserstein gradient flow structure of the Fokker-Planck equation and...
In this talk, I will present a new asymptotic preserving scheme for kinetic equations of Boltzmann-BGK type in the diffusive scaling. The scheme is a suitable combination of micro-macro decomposition, the micro part being discretized by a particle method, and Monte Carlo techniques. Thanks to the Monte Carlo particle approximation, the computational cost of the method automatically reduces...
In this work we consider PDE models used in plasmas physic like MHD or Vlasov equations.
The key point to solve the kinetic Vlasov equation is the Semi Lagrangian Solver, which is a high-order, CFL less and Matrix-free solver for the transport equation.
The other models present in plasma physic like MHD, anisotropic equation or Poisson solver can be written like approximated BGK models.
For...
The Fisher-KPP equation is a diffusion equation with logistic reaction modeling the time evolution of the density of one species confined in the bounded domain.
According to this interpretation, we expect that the density remains non-negative during the evolution. Despite in the continuous setting it is not difficult to prove this, at the discrete level the same results are not trivial at...
