Related papers: On absence of embedded eigenvalues and stability o…
We perform a complete spectral analysis of the linear three-dimensional Boltzmann BGK operator resulting in an explicit transcendental equation for the eigenvalues. Using the theory of finite-rank perturbations, we confirm the existence of…
We study the linearized Vlasov equations and the linearized Vlasov-Fokker-Planck equations in the weakly collisional limit in a uniform magnetic field. In both cases, we consider periodic confinement and Maxwellian (or close to Maxwellian)…
We prove nonlinear Landau damping in optimal weighted Gevrey-3 spaces for solutions of the confined Vlasov-Poisson system on $\T^d\times\R^d$ which are small perturbations of homogeneous Penrose-stable equilibria. We also prove the…
In order to understand the nonlinear stability of many types of time-periodic travelling waves on unbounded domains, one must overcome two main difficulties: the presence of embedded neutral eigenvalues and the time-dependence of the…
We prove asymptotic stability of the Poisson homogeneous equilibrium among solutions of the Vlassov-Poisson system in the Euclidean space $\mathbb{R}^3$. More precisely, we show that small, smooth, and localized perturbations of the Poisson…
A common challenge to proving asymptotic stability of solitary waves is understanding the spectrum of the operator associated with the linearized flow. The existence of eigenvalues can inhibit the dispersive estimates key to proving…
We study certain stationary and time-evolution problems of trapped Bose-Einstein condensates of weakly interacting alkali atoms described by a nonlinear Gross-Pitaevskii (GP) equation. We suggest a pseudospectral method involving Laguerre…
This review presents an upgraded wave theory adapted to the high fluctuation level of driven realistic i.e. non-idealized plasmas. Above all, this means giving up the well-known concept of a linear wave theory in favor of a thoroughly…
The longitudinal dynamics of an intense high energy beam moving in a resonator cavity has been studied in some detail. Through the method of separation of variables and its obvious straightforward generalization, a solution of the Vlasov…
The aim of this article is to prove that the linear stability or instability of small Bernstein-Green-Kruskal (BGK) waves is determined by the sign of the derivative of their energy distributions at $0$ energy.
The Bogoliubov-de Gennes equations are used for a number of theoretical works on the trapped Bose-Einstein condensates. These equations are known to give the energies of the quasi-particles when all the eigenvalues are real. We consider the…
We study the Bathnagar-Gross-Krook (BGK) equation in a smooth bounded domain featuring a diffusive reflection boundary condition with general collision frequency. We prove that the BGK equation admits a unique global solution with an…
We study hypocoercivity for a class of linearized BGK models for continuous phase spaces. We develop methods for constructing entropy functionals that enable us to prove exponential relaxation to equilibrium with explicit and physically…
A nonlinear Langmuir wave in the kinetic regime $k\lambda_D\gtrsim0.2$ may have a filamentation instability, where $k$ is the wavenumber and $\lambda_D$ is the Debye length. The nonlinear stage of that instability develops into the…
We study the numerical resolution of the time-dependent Gross-Pitaevskii equation, a non-linear Schroedinger equation used to simulate the dynamics of Bose-Einstein condensates. Considering condensates trapped in harmonic potentials, we…
We study the non-linear BGK model in 1d coupled to a spatially varying thermostat. We show existence, local uniqueness and linear stability of a steady state when the linear coupling term is large compared to the non-linear self interaction…
In this paper, we establish the large time asymptotic behavior of solutions to the linearized Vlasov-Poisson system near general spatially homogenous equilibria $\mu(\frac12|v|^2)$ with connected support on the torus $\mathbb{T}^3_x \times…
The electron energy distribution function (EEDF) in low-temperature plasmas exhibits features not fully captured by classical collisional models, particularly across the transition from kinetic to hydrodynamic regimes. This work attributes…
A discontinuous Galerkin method for approximating the Vlasov-Poisson system of equations describing the time evolution of a collisionless plasma is proposed. The method is mass conservative and, in the case that piecewise constant functions…
We prove the asymptotic stability of standing kink for the nonlinear relativistic wave equations of the Ginzburg-Landau type in one space dimension: for any odd initial condition in a small neighborhood of the kink, the solution,…