Detailed scope

 Kinetic and fluid equations provide important insights in plasma physics and gravitational dynamics, and pose important problems in analysis and numerical analysis. This workshop will focus on the articulation between these descriptions.

Kinetic limits in the context of Hamiltonian or diffusive/dissipative dynamics

This involves showing that a system composed of a large number N of particles, governed by a distribution function (BBGKY hierarchy) or an empirical measure in phase-space, can be described by a kinetic equation with a one-particle distribution function (even also sometimes with one- and two-particle distribution functions coupled together) when N tends to infinity. These particle systems can be Hamiltonian or dissipative/diffusive. For Hamiltonian systems, one can mention the kinetic limit of a system subjected to the Coulomb or Newtonian force towards the Vlasov-Poisson equation. In astrophysics, one can think about stars in galaxies or dark matter in the Universe. For diffusive systems, one can cite the kinetic limit of particles subjected to billiard dynamics (ballistic transport plus elastic shocks) to the classical collisional Boltzmann equation. One can also mention other more complex kinetic limits such as that towards the Balescu-Lenard-Guernsey equation.

Fluid limits

It is a question of showing that in certain physical regimes where some dimension-free parameters become small, a kinetic equation can be approximated by fluid equations on the moments of the distribution function in velocity space. In the dissipative case, one can quote the approximation of the collisional Boltzmann equation by the Navier-Stokes equations when the mean free path between two collisions is small compared to the size of the system (or small Knudsen number). In the Hamiltonian case, one can, for example, cite the Vlasov-Poisson fluid limit to the incompressible Euler equations in the quasi-neutral limit.

Approximate models and applications

The collective behavior in these models gives rise to waves, which can also be considered as dynamic actors. Wave-particle and wave-wave interactions are physico-mathematical mechanisms at the base of many crucial approximate models in physics. For example, for the linear wave-particle interaction, when the phase velocity of a wave is close to the velocity of a particle, the quasi-linear approximation of the Vlasov equation accounts for the saturation of electromagnetic instabilities by a velocity diffusion equation on the spatial mean of the distribution function. Concerning the wave-wave interaction, one can quote weak turbulence, which makes it possible to predict the behavior of the wave spectrum in fluid mechanics by modelling the transfer of energy between large and small scales due to the coupling of waves produced by non-linear terms. Analogous approaches allow one to predict the evolution of the power-spectrum of the large-scale matter distribution in the Universe with higher order cosmological perturbation theory. In quantum mechanics, one can also think about the approximation of the Schrödinger equation by a kinetic equation of waves of the collisional Boltzmann type. Finally,  propagation of chaos is a cross-cutting theme to the previous ones. In this case, it is a question of proving that, if a system of particles initially satisfies certain statistical properties (e.g., independence), then they are conserved over time by the dynamics, justifying certain reduced descriptions.