Existence and non-uniqueness of stationary states for the Vlasov-Poisson equation on \(\mathbb{R}^3\) subject to attractive background charges. (English) Zbl 1516.35415
Summary: We prove the existence of stationary solutions for the density of an infinitely extended plasma interacting with an arbitrary configuration of background charges. Furthermore, we show that the solution cannot be unique if the total charge of the background is attractive. In this case, infinitely many different stationary solutions exist. The non-uniqueness can be explained by the presence of trapped particles orbiting the attractive background charge.
MSC:
| 35Q83 | Vlasov equations |
References:
| [1] | Arroyo-Rabasa, A.; Winter, R., Debye screening for the stationary Vlasov-Poisson equation in interaction with a point charge, Commun. Partial Differ. Equ., 46, 8, 1569-1584 (2021) · Zbl 1484.82004 · doi:10.1080/03605302.2021.1892754 |
| [2] | Duan, R.; Strain, R., Optimal time decay of the Vlasov-Poisson-Boltzmann system in \(\mathbb{R}^3\), Arch Ration. Mech. Anal., 199, 1, 291-328 (2011) · Zbl 1232.35169 · doi:10.1007/s00205-010-0318-6 |
| [3] | Duan, R.; Yang, T., Stability of the one-species Vlasov-Poisson-Boltzmann system, SIAM J. Math. Anal., 41, 6, 2353-2387 (2010) · Zbl 1323.82036 · doi:10.1137/090745775 |
| [4] | Duan, R.; Yang, T.; Zhu, C., Existence of stationary solutions to the Vlasov-Poisson-Boltzmann system, J. Math. Anal. Appl., 327, 1, 425-434 (2007) · Zbl 1253.82079 · doi:10.1016/j.jmaa.2006.04.047 |
| [5] | Goldston, R.; Rutherford, P., Introduction to Plasma Physics (1995), Boca Raton: CRC Press, Boca Raton · doi:10.1201/9781439822074 |
| [6] | Nicholson, D., Introduction to Plasma Theory (1983), New York: Wiley, New York |
| [7] | Pausader, B.; Widmayer, K., Stability of a point charge for the Vlasov-Poisson system: the radial case, Commun. Math. Phys., 385, 3, 1741-1769 (2021) · Zbl 1475.35343 · doi:10.1007/s00220-021-04117-8 |
| [8] | Pausader, B., Widmayer, K., Yang, J.: Stability of a point charge for the repulsive Vlasov-Poisson system. arXiv:2207.05644 (2022) |
| [9] | Rein, G., Non-linear stability for the Vlasov-Poisson system-the energy-Casimir method, Math. Methods Appl. Sci., 17, 14, 1129-1140 (1994) · Zbl 0814.76094 · doi:10.1002/mma.1670171404 |
| [10] | Schamel, H., Stationary solutions of the electrostatic Vlasov equation, Plasma Phys., 13, 6, 491-505 (1971) · doi:10.1088/0032-1028/13/6/005 |
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