Stability of the stationary solution to VlasovPoissonBoltzmann system with hard potential
帶硬位勢的 VlasovPoissonBoltzmann 系統靜態解的穏定性
Student thesis: Doctoral Thesis
Author(s)
Related Research Unit(s)
Detail(s)
Awarding Institution  

Supervisors/Advisors 

Award date  15 Feb 2012 
Link(s)
Permanent Link  https://scholars.cityu.edu.hk/en/theses/theses(b496485642764ff9bc85ae93094d436b).html 

Other link(s)  Links 
Abstract
The VlasovPoissonBoltzmann (VPB) system is a physical model describing
the time evolution of number density distribution of charged dilute gases.
The selfconsistent potential generating the electric field is related to the
unknown distribution function by the Poisson equation. There is extensive
research on the VPB system and some new phenomena in the long time
behaviour have been discovered.
However, most of the previous research on the Cauchy problem is about
the hard sphere case, and the difficulty of extending the results to the hard
potential case lies in controlling the largevelocity growth in the nonlinear
term. Recently, there are some studies about this hard potential case with
constant background density and the analysis is based on introducing a new
timevelocity weight function in the energy functional and its dissipation
to control the large growth. In this thesis, we consider the case for hard
potential with stationary background density.
The commonly used perturbation scheme leads to an equation with a term
involving no dissipation. Hence, it's very difficult to control. We deal with
this term by introducing a new perturbation of the stationary solution. Then
by multiplying the collision invariants, we derive the fluidtype equations for
the macroscopic parts of the solution. Based on the resulting equations and
an a priori assumption about the decay rate of the selfconsistent electric
potential, we obtain the energy estimates for the solution and its dissipation
rate given by the microscopic parts. We also need the macroscopic dissipation
rate to absorb the right hand terms appearing in the previous estimates. To
do this, we show the existence of an interactive energy functional and its
dissipation rate contains macroscopic parts of the solution.
In order to absorb one term arising from the a priori assumption in the
above estimates, we need the weighted energy estimates. Using induction
and the linear combination of the previous nonweighted energy estimates,
we obtain the desired estimates and its dissipation given by the full weighted
dissipation rate.
Subsequently, we show the local existence of the unique solution under
the natural mass conservation condition and two a priori assumptions. The
analysis is based on the previously obtained weighted energy estimates, the
high order energy estimates and time decay estimate for a linearized system.
Finally, we obtain the uniform in time estimates of the solution by the continuity
argument and show that the a priori assumptions we made can be
closed.
 Mathematical models, Transport theory, Kinetic theory of gases