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The Vlasov-Poisson-Boltzmann system near Maxwellians. (English) Zbl 1027.82035

This paper deals with the dynamics of dilute eletrons which can be modelled by the Vlasov-Poisson-Boltzmann system:

t F+v· x F+ x ϕ· v F=Q[F,F]Δϕ=ρ-ρ 0 = 3 Fdv-ρ 0 ; 𝕋 3 ϕdx=0F(0,x,v)=F 0 (x,v),(1)

where F(t,x,v) is the spatially periodic distribution function for the particles at time t0, spatial coordinates x=(x 1 ,x 2 ,x 3 )[-π,π] 3 =𝕋 3 , and velocity v=(v 1 ,v 2 ,v 3 ). It is shown that any smooth, periodic initial perturbation of a given global Maxwellian that preserves the same mass, momentum, and total energy, leads to a unique global-in-time classical solution. The construction of global solutions is based on an energy method.


MSC:
82C40Kinetic theory of gases (time-dependent statistical mechanics)
35A05General existence and uniqueness theorems (PDE) (MSC2000)
35L60Nonlinear first-order hyperbolic equations
35Q60PDEs in connection with optics and electromagnetic theory
45K05Integro-partial differential equations
82D10Plasmas (statistical mechanics)