Decay of the two-species Vlasov–Poisson–Boltzmann system

We establish the time decay rates of the solution to the Cauchy problem for the two-species Vlasov–Poisson–Boltzmann system near Maxwellians via a refined pure energy method. The total density of two species of particles decays at the optimal algebraic rate as the Boltzmann equation, but the disparity between two species and the electric field decay at an exponential rate. This phenomenon reveals the essential difference when compared to the one-species Vlasov–Poisson–Boltzmann system or the Navier–Stokes–Poisson equations in which the electric field decays at the optimal algebraic rate, and compared to the Vlasov–Boltzmann system in which the disparity between two species decays at the optimal algebraic rate. Our achievement heavily relies on a reformulation of the problem which well displays the cancelation property of the two-species system, and our proof is based on a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis.     

Negative Sobolev spaces and the two-species Vlasov–Maxwell–Landau system in the whole space

A global solvability result of the Cauchy problem of the two-species Vlasov–Maxwell–Landau system near a given global Maxwellian is established by employing an approach different than that of [2]. Compared with that of [2], the minimal regularity index and the smallness assumptions we imposed on the initial data are weaker. Our analysis does not rely on the decay of the corresponding linearized system and the Duhamel principle and thus it can be used to treat the one-species Vlasov–Maxwell–Landau system for the case of γ>−3$\gamma >-3$ and the one-species Vlasov–Maxwell–Boltzmann system for the case of −1<γ≤1$-1<\gamma \le 1$ to deduce the global existence results together with the corresponding temporal decay estimates.     

Stability of rarefaction waves for the non-cutoff Vlasov–Poisson–Boltzmann system with physical boundary

In this paper, we are concerned with the Vlasov–Poisson–Boltzmann (VPB) system in three-dimensional spatial space without angular cutoff in a rectangular duct with or without physical boundary conditions. Near a local Maxwellian with macroscopic quantities given by rarefaction wave solution of one-dimensional compressible Euler equations, we establish the time-asymptotic stability of planar rarefaction wave solutions for the Cauchy problem to VPB system with periodic or specular-reflection boundary condition. In particular, we successfully introduce physical boundaries, namely, specular-reflection boundary, to the models describing wave patterns of kinetic equations. Moreover, we treat the non-cutoff collision kernel instead of the cutoff one. As a simplified model, we also consider the stability and large time behavior of the rarefaction wave solution for the Boltzmann equation.     

Global existence of classical solutions to the Vlasov–Poisson–Boltzmann system with given magnetic field

This paper considers the Vlasov–Poisson–Boltzmann system with given magnetic field. The global existence of classical solutions was obtained when the initial data is a small perturbation around a global Maxwellian. The proof is based on the theory of compressible Navier–Stokes–Poisson equations with forcing and the macro–microdecomposition of the solution to the Boltzmann equation with respect to the local Maxwellian introduced in [T.-P. Liu, T. Yang, S.-H. Yu, Energy method for the Boltzmann equation, Physica D 188 (3–4) (2004) 178–192] and elaborated in [T. Yang, H.-J. Zhao, A new energy method for the Boltzmann equation, J. Math. Phys. 47 (2006)]. The result shows that the existence of solutions is independent of the magnetic field.     

Vlasov–Poisson: The waterbag method revisited

We revisit, with a view to refinement and generalization, the elegant waterbag method for the numerical treatment of Vlasov–Poisson equations. In this method, the phase space is decomposed into patches of constant density, and by exploiting Liouville’s theorem, the dynamics is reduced to the evolution of the boundary of these patches (waterbags). We follow the boundary using an adaptive, oriented polygon, and recover the force by circulating along this polygon. We discuss sampling of initial conditions with a set of oriented isocontours, and propose a new refinement procedure for accurate rendering of the stretching and folding polygon. Time evolution is naturally undertaken with symplectic algorithms. Tools, initially developed for systems of self-gravitating sheets, generalize naturally to spherically symmetric systems. We conclude with examples of both cases.     

Solutions of the Vlasov–Maxwell–Boltzmann system with long-range interactions

We establish the existence of renormalized solutions of the Vlasov–Maxwell–Boltzmann system with a defect measure in the presence of long-range interactions. We also present a control of the defect measure by the entropy dissipation only, which turns out to be crucial in the study of hydrodynamic limits.     

A generalized Fourier–Hermite method for the Vlasov–Poisson system

BIT Numerical Mathematics - A generalized Fourier–Hermite semi-discretization for the Vlasov–Poisson equation is introduced. The formulation of the method includes as special cases the...     

Quasi-neutral limit for the full Euler–Poisson system in one-dimensional space

In this paper, we consider the quasi-neutral limit of the full Euler–Poisson system in one-dimensional space when the Debye length tends to zero. Due to the observation that the full Euler–Poisson system is Friedrich symmetrizable, we can obtain uniform estimates by applying the pseudo-differential energy estimates. It is shown that for well-prepared initial data the strong solution of the full Euler–Poisson system converges strongly to the compressible Euler equations in small time interval.     

From the Vlasov–Poisson equation with strong local alignment to the pressureless Euler–Poisson system

This article provides a rigorous justification on a hydrodynamic limit from the Vlasov–Poisson system with strong local alignment to the pressureless Euler–Poisson system for repulsive dynamics.     

On the controllability of the Vlasov–Poisson system in the presence of external force fields

In this work, we are interested in the controllability of Vlasov–Poisson systems in the presence of an external force field (namely a bounded force field or a magnetic field), by means of a local interior control. We are able to extend the results of Glass (2003) [8], where the only present force was the self-consistent electric field.     

Global weak solutions to the Vlasov–Poisson–Fokker–Planck–Navier–Stokes system

We consider the compressible Vlasov–Poisson–Fokker–Planck–Navier–Stokes system in a three-dimensional bounded domain with nonhomogeneous Dirichlet boundary conditions. The system describes the evolution of charged particles ensemble dispersed in an isentropic fluid. For the adiabatic coefficient \" data-semantic-complexity=\"1\">, we establish the global existence of weak solutions to this system with arbitrary large initial and boundary data.