The focusing Continuum Calogero–Moser (CCM) equation is a completely integrable PDE that describes a continuum limit of a particle gas interacting pairwise through an inverse square potential. This system is well-posed in the scaling-critical space L2 below the mass of the soliton, but above this threshold there are solutions that blow up in finite time. In this talk, we will discuss some new and existing results about solutions below the soliton mass threshold. This is based on joint work with Rowan Killip and Monica Visan.[-]

Suspensions are ubiquitous in nature (sediments, clouds,biological fluids ... etc.) and in industry such as civil engineering (paints, polymers ... etc.) among many others. The rigorous derivation of fluid-kinetic models for suspensions has attracted a lot of attention in the last decade. This lecture aims at presenting a review of the main results that have been obtained.

The first session aims at introducing both the microscopic and the limiting equation and giving a formal derivation of the former one. The second session aims at presenting the main early results concerning the derivation of an effective model starting from the microscopic model in which particle positions and velocities are fixed or given. Such a system takes the following form for example
\begin{equation}\label{eq:Stokes}
\left \{
\begin{array}{rcl}
-\Delta u+\nabla p &=& f, \text{ on } \Omega\setminus \overline{\underset{i=1}{\overset{N}{\bigcup}} B(x_i,r)} \\
\text{div } u&=& 0, \text{ on } \Omega\setminus \overline{\underset{i=1}{\overset{N}{\bigcup}} B(x_i,r)} \\
u&=& V_i, \text{ on } \partial B(x_i,r)\\
u&=& 0, \text{ on } \partial \Omega
\end{array}
\right.
\end{equation}
where $\Omega$ a smooth open set of $\mathbb{R}^3$, $x_1, x_2, \cdots, x_N$ are the particles position, $r$ their radius and $V_i$ the given velocity of the $i$th particle. The aim is then to perform an asymptotic analysis when the number of particles $N$ becomes large while their radius $r$ becomes small, first results have been obtained in [1,2,3] where the limit equations depend on the scale of the holes and their typical distance; Stokes equation, Darcy equation or Stokes-Brinkman equation. After recalling the recent contributions, we will present a short argument giving insights about the derivation of the Brinkman term in a simple case.

The last session of this mini-course aims at presenting the results regarding the rigorous derivation of fluid-kinetic models when taking into account the fluid-particle interactions and particle dynamics. This means that we consider the Stokes equation [1] coupled to Newton laws where we neglect particles inertia (balance of force and torque) and the motion of the center of the particles $\dot{x}_i=V_i$.

The rigorous derivation of a fluid-kinetic model in this setting have been obtained in [6,5,7] in the case $\Omega=\mathbb{R}^3$ under some separation assumptions on the particles. The obtained equation is a Transport-Stokes equation
\begin{equation}\label{eq:TS}\tag{TS}
\left\{
\begin{array}{rcl}
- \Delta u + \nabla p &=& \rho g,\\
\text{div } u&=& 0, \\
\partial_t \rho +\text{div }( ( u + \gamma^{-1} V_{\mathrm{St}})\rho) &=& 0,
\end{array}
\right.
\end{equation}
where $\gamma = \lim Nr \in (0,\infty]$.

This result is related to the mean field limit of many particles interacting through a kernel and has been extensively studied for several different problems. We present the main ideas for such a derivation using the method of reflections and stability estimates through Wasserstein distance following the approach by M. Hauray [4]. We finish by emphasizing new results based on a mean-field argument for the derivation of models of suspensions.[-]

We investigate the mean-field limit of large networks of interacting biological neurons. The neurons are represented by the so-called integrate and fire models that follow the membrane potential of each neuron and captures individual spikes. However we do not assume any structure on the graph of interactions but consider instead any connection weights between neurons that obey a generic mean-field scaling. We are able to extend the concept of extended graphons, introduced in Jabin-Poyato-Soler, by introducing a novel notion of discrete observables in the system. This is a joint work with D. Zhou.[-]

We investigate the mean-field limit of large networks of interacting biological neurons. The neurons are represented by the so-called integrate and fire models that follow the membrane potential of each neuron and captures individual spikes. However we do not assume any structure on the graph of interactions but consider instead any connection weights between neurons that obey a generic mean-field scaling. We are able to extend the concept of extended graphons, introduced in Jabin-Poyato-Soler, by introducing a novel notion of discrete observables in the system. This is a joint work with D. Zhou.[-]

Suspensions are ubiquitous in nature (sediments, clouds,biological fluids ... etc.) and in industry such as civil engineering (paints, polymers ... etc.) among many others. The rigorous derivation of fluid-kinetic models for suspensions has attracted a lot of attention in the last decade. This lecture aims at presenting a review of the main results that have been obtained.

The first session aims at introducing both the microscopic and the limiting equation and giving a formal derivation of the former one. The second session aims at presenting the main early results concerning the derivation of an effective model starting from the microscopic model in which particle positions and velocities are fixed or given. Such a system takes the following form for example
\begin{equation}\label{eq:Stokes}
\left \{
\begin{array}{rcl}
-\Delta u+\nabla p &=& f, \text{ on } \Omega\setminus \overline{\underset{i=1}{\overset{N}{\bigcup}} B(x_i,r)} \\
\text{div } u&=& 0, \text{ on } \Omega\setminus \overline{\underset{i=1}{\overset{N}{\bigcup}} B(x_i,r)} \\
u&=& V_i, \text{ on } \partial B(x_i,r)\\
u&=& 0, \text{ on } \partial \Omega
\end{array}
\right.
\end{equation}
where $\Omega$ a smooth open set of $\mathbb{R}^3$, $x_1, x_2, \cdots, x_N$ are the particles position, $r$ their radius and $V_i$ the given velocity of the $i$th particle. The aim is then to perform an asymptotic analysis when the number of particles $N$ becomes large while their radius $r$ becomes small, first results have been obtained in [1,2,3] where the limit equations depend on the scale of the holes and their typical distance; Stokes equation, Darcy equation or Stokes-Brinkman equation. After recalling the recent contributions, we will present a short argument giving insights about the derivation of the Brinkman term in a simple case.

The last session of this mini-course aims at presenting the results regarding the rigorous derivation of fluid-kinetic models when taking into account the fluid-particle interactions and particle dynamics. This means that we consider the Stokes equation [1] coupled to Newton laws where we neglect particles inertia (balance of force and torque) and the motion of the center of the particles $\dot{x}_i=V_i$.

The rigorous derivation of a fluid-kinetic model in this setting have been obtained in [6,5,7] in the case $\Omega=\mathbb{R}^3$ under some separation assumptions on the particles. The obtained equation is a Transport-Stokes equation
\begin{equation}\label{eq:TS}\tag{TS}
\left\{
\begin{array}{rcl}
- \Delta u + \nabla p &=& \rho g,\\
\text{div } u&=& 0, \\
\partial_t \rho +\text{div }( ( u + \gamma^{-1} V_{\mathrm{St}})\rho) &=& 0,
\end{array}
\right.
\end{equation}
where $\gamma = \lim Nr \in (0,\infty]$.

This result is related to the mean field limit of many particles interacting through a kernel and has been extensively studied for several different problems. We present the main ideas for such a derivation using the method of reflections and stability estimates through Wasserstein distance following the approach by M. Hauray [4]. We finish by emphasizing new results based on a mean-field argument for the derivation of models of suspensions.


Several extensions have been made, we mention for instance [3] where authors considered steady Navier Stokes equation with non periodically distributed particles satisfying a minimal distance assumption and for general Dirichlet boundary conditions with uniform kinetic energy, they in particular characterized the convergence in terms of the limit of (marginals of) the empirical measure

$\rho^N(x)=\frac{1}{N} \underset{1 \leq i \leq n}{\sum} \delta_{x_i}
$

Several extensions have been then obtained. We mention [5] where the author extends the minimal distance assumption, for quantitative convergence estimates, [6] in the case of arbitrary shaped particles, the case of randomly distributed particles, fora simplified proof.
[-]

Suspensions are ubiquitous in nature (sediments, clouds,biological fluids ... etc.) and in industry such as civil engineering (paints, polymers ... etc.) among many others. The rigorous derivation of fluid-kinetic models for suspensions has attracted a lot of attention in the last decade. This lecture aims at presenting a review of the main results that have been obtained.

The first session aims at introducing both the microscopic and the limiting equation and giving a formal derivation of the former one. The second session aims at presenting the main early results concerning the derivation of an effective model starting from the microscopic model in which particle positions and velocities are fixed or given. Such a system takes the following form for example
\begin{equation}\label{eq:Stokes}
\left \{
\begin{array}{rcl}
-\Delta u+\nabla p &=& f, \text{ on } \Omega\setminus \overline{\underset{i=1}{\overset{N}{\bigcup}} B(x_i,r)} \\
\text{div } u&=& 0, \text{ on } \Omega\setminus \overline{\underset{i=1}{\overset{N}{\bigcup}} B(x_i,r)} \\
u&=& V_i, \text{ on } \partial B(x_i,r)\\
u&=& 0, \text{ on } \partial \Omega
\end{array}
\right.
\end{equation}
where $\Omega$ a smooth open set of $\mathbb{R}^3$, $x_1, x_2, \cdots, x_N$ are the particles position, $r$ their radius and $V_i$ the given velocity of the $i$th particle. The aim is then to perform an asymptotic analysis when the number of particles $N$ becomes large while their radius $r$ becomes small, first results have been obtained in [1,2,3] where the limit equations depend on the scale of the holes and their typical distance; Stokes equation, Darcy equation or Stokes-Brinkman equation. After recalling the recent contributions, we will present a short argument giving insights about the derivation of the Brinkman term in a simple case.

The last session of this mini-course aims at presenting the results regarding the rigorous derivation of fluid-kinetic models when taking into account the fluid-particle interactions and particle dynamics. This means that we consider the Stokes equation [1] coupled to Newton laws where we neglect particles inertia (balance of force and torque) and the motion of the center of the particles $\dot{x}_i=V_i$.

The rigorous derivation of a fluid-kinetic model in this setting have been obtained in [6,5,7] in the case $\Omega=\mathbb{R}^3$ under some separation assumptions on the particles. The obtained equation is a Transport-Stokes equation
\begin{equation}\label{eq:TS}\tag{TS}
\left\{
\begin{array}{rcl}
- \Delta u + \nabla p &=& \rho g,\\
\text{div } u&=& 0, \\
\partial_t \rho +\text{div }( ( u + \gamma^{-1} V_{\mathrm{St}})\rho) &=& 0,
\end{array}
\right.
\end{equation}
where $\gamma = \lim Nr \in (0,\infty]$.

This result is related to the mean field limit of many particles interacting through a kernel and has been extensively studied for several different problems. We present the main ideas for such a derivation using the method of reflections and stability estimates through Wasserstein distance following the approach by M. Hauray [4]. We finish by emphasizing new results based on a mean-field argument for the derivation of models of suspensions.[-]

This lecture is devoted to the characterization of convergence rates in some simple equations with mean field nonlinear couplings, like the Keller-Segel and Nernst-Planck systems, Cucker-Smale type models, and the Vlasov-Poisson-Fokker-Planck equation. The key point is the use of Lyapunov functionals adapted to the nonlinear version of the model to produce a functional framework adapted to the asymptotic regime and the corresponding spectral analysis.[-]

We propose a modulated free energy which combines of the method previously developed by the speaker together with the modulated energy introduced by S. Serfaty. This modulated free energy may be understood as introducing appropriate weights in the relative entropy to cancel the more singular terms involving the divergence of the flow. This modulated free energy allows to treat singular interactions of gradient-flow type and allows potentials with large smooth part, small attractive singular part and large repulsive singular part. As an example, a full rigorous derivation (with quantitative estimates) of some chemotaxis models, such as Patlak-Keller Segel system in the subcritical regimes, is obtained. This is joint work with D. Bresch and Z. Wang.[-]

We propose a modulated free energy which combines of the method previously developed by the speaker together with the modulated energy introduced by S. Serfaty. This modulated free energy may be understood as introducing appropriate weights in the relative entropy to cancel the more singular terms involving the divergence of the flow. This modulated free energy allows to treat singular interactions of gradient-flow type and allows potentials with large smooth part, small attractive singular part and large
repulsive singular part. As an example, a full rigorous derivation (with quantitative estimates) of some chemotaxis models, such as Patlak-Keller Segel system in the subcritical regimes, is obtained. This is joint work with D. Bresch and Z. Wang.[-]

Nonlocal interaction energies are continuum models for large systems of particles, where typically each particle interacts not only with its immediate neighbors, but also with particles that are far away. Examples of these energies arise in many different applications, such as biology (population dynamics), physics (Ginzburg-Landau vortices), and material science (dislocation theory). A fundamental question is understanding the optimal arrangement of particles at equilibrium, which are described, at least in average, by minimizers of the energy. In this talk I will focus on a class of nonlocal energies that are perturbations of the Coulomb energy and I will show how their minimizers can be explicitly characterized. This is based on joint works with J. Mateu, L. Rondi, L. Scardia, and J. Verdera.[-]