Daniel Han-Kwan

Quasineutral limit of the Vlasov-Poisson system with massless electrons

In this paper, we study the quasineutral limit (in other words the limit when the Debye length tends to zero) of Vlasov-Poisson like equations describing the behavior of ions in a plasma. We consider massless electrons, with a charge density following a Maxwell-Boltzmann law. For cold ions, using the relative entropy method, we derive the classical Isothermal Euler or the (inviscid) Shallow Water systems from fluid mechanics. We then study the combined quasineutral and strong magnetic field regime for such plasmas.

Stability Issues in the Quasineutral Limit of the One-Dimensional Vlasov–Poisson Equation

This work is concerned with the quasineutral limit of the one-dimensional Vlasov–Poisson equation, for initial data close to stationary homogeneous profiles. Our objective is threefold: first, we provide a proof of the fact that the formal limit does not hold for homogeneous profiles that satisfy the Penrose instability criterion. Second, we prove on the other hand that the limit is true for homogeneous profiles that satisfy some monotonicity condition, together with a symmetry condition. We handle the case of well-prepared as well as ill-prepared data. Last, we study a stationary boundary-value problem for the formal limit, the so-called quasineutral Vlasov equation. We show the existence of numerous stationary states, with a lot of freedom in the construction (compared to that of BGK waves for Vlasov–Poisson): this illustrates the degeneracy of the limit equation.

The three-dimensional finite Larmor radius approximation

Following Frenod and Sonnendrucker (SIAM J. Math. Anal. 32(6) (2001) 1227-1247), we consider the finite Larmor radius regime for a plasma submitted to a large magnetic field and take into account both the quasineutrality and the local thermodynamic equilibrium of the electrons. We then rigorously establish the asymptotic gyrokinetic limit of the rescaled and modified Vlasov-Poisson system in a three-dimensional setting with the help of an averaging lemma.

From Vlasov-Poisson to Korteweg-de Vries and Zakharov-Kuznetsov

We introduce a long wave scaling for the Vlasov–Poisson equation and derive, in the cold ions limit, the Korteweg–de Vries equation (in 1D) and the Zakharov–Kuznetsov equation (in higher dimensions, in the presence of an external magnetic field). The proofs are based on the relative entropy method.

On the confinement of a tokamak plasma

The goal of this paper is to understand from a mathematical point of view the magnetic confinement of plasmas for fusion. Following Frenod and Sonnendrucker [SIAM J. Math. Anal., 32 (2001), pp. 1227–1247], we first use two-scale convergence tools to derive a gyrokinetic system for a plasma submitted to a large magnetic field with a slowly spatially varying intensity. We formally derive from this system a simplified bitemperature fluid system. We then investigate the behavior of the plasma in such a regime, and we prove nonlinear stability or instability depending on which side of the tokamak we are looking at. In our analysis, we will also point out that there exists a temperature gradient threshold beyond which one can expect stability, even in the “bad” side: this corresponds to the so-called H-mode.

Nonlinear Instability of Vlasov--Maxwell Systems in the Classical and Quasineutral Limits

We study the instability of solutions to the relativistic Vlasov--Maxwell systems in two limiting regimes: the classical limit when the speed of light tends to infinity and the quasineutral limit when the Debye length tends to zero. First, in the classical limit,

Instabilities in the Mean Field Limit

\varepsilon \to 0

Long Time Estimates for the Vlasov–Maxwell System in the Non-relativistic Limit

, with

Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries

\varepsilon

Quasineutral limit for Vlasov-Poisson with Penrose stable data

being the inverse of the speed of light, we construct a family of solutions that converge initially polynomially fast to a homogeneous solution