Existence of stationary solutions to the Vlasov-Poisson-Boltzmann system. (English) Zbl 1253.82079
Summary: In this paper, we study the existence of stationary solutions to the Vlasov-Poisson-Boltzmann system when the background density function tends to a positive constant with a very mild decay rate as \(| x|\to\infty\). In fact, the stationary Vlasov-Poisson-Boltzmann system can be written into an elliptic equation with exponential nonlinearity. Under the assumption on the decay rate being \((\ln(e+| x|))-\alpha\) for some \(\alpha>0\), it is shown that this elliptic equation has a unique solution. This result generalizes the previous work [R. Glassey, J. Schaeffer and Y. Zheng, J. Math. Anal. Appl. 202, No. 3, 1058–1075 (1996; Zbl 0867.35026)] where the decay rate \((1+| x|)^{-1/2}\) is assumed.
MSC:
| 82D10 | Statistical mechanics of plasmas |
| 35Q83 | Vlasov equations |
| 35J60 | Nonlinear elliptic equations |
| 76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |
| 76X05 | Ionized gas flow in electromagnetic fields; plasmic flow |
| 82C40 | Kinetic theory of gases in time-dependent statistical mechanics |
Citations:
Zbl 0867.35026References:
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