Stability of the stationary solution to Vlasov-Poisson-Boltzmann system with hard potential
帶硬位勢的 Vlasov-Poisson-Boltzmann 系統靜態解的穏定性
Student thesis: Doctoral Thesis
Related Research Unit(s)
The Vlasov-Poisson-Boltzmann (VPB) system is a physical model describing the time evolution of number density distribution of charged dilute gases. The self-consistent 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 large-velocity 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 time-velocity 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 fluid-type equations for the macroscopic parts of the solution. Based on the resulting equations and an a priori assumption about the decay rate of the self-consistent 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 non-weighted 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