On the magnetic shield for a Vlasov-Poisson plasma. (English) Zbl 1387.35577
Summary: We study the screening of a bounded body \(\Gamma \) against the effect of a wind of charged particles, by means of a shield produced by a magnetic field which becomes infinite on the border of \(\Gamma \). The charged wind is modeled by a Vlasov-Poisson plasma, the bounded body by a torus, and the external magnetic field is taken close to the border of \(\Gamma \). We study two models: a plasma composed by different species with positive or negative charges, and finite total mass of each species, and another made of many species of the same sign, each having infinite mass. We investigate the time evolution of both systems, showing in particular that the plasma particles cannot reach the body. Finally we discuss possible extensions to more general initial data. We show also that when the magnetic lines are straight lines, (that imposes an unbounded body), the previous results can be improved.
MSC:
| 35Q83 | Vlasov equations |
| 82D10 | Statistical mechanics of plasmas |
| 76X05 | Ionized gas flow in electromagnetic fields; plasmic flow |
| 35L60 | First-order nonlinear hyperbolic equations |
References:
| [1] | Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1970) · Zbl 0958.76001 |
| [2] | Caglioti, E., Caprino, S., Marchioro, C., Pulvirenti, M.: The Vlasov equation with infinite mass. Arch. Ration. Mech. Anal. 159, 85-108 (2001) · Zbl 1021.76059 · doi:10.1007/s002050100150 |
| [3] | Caprino, S., Cavallaro, G., Marchioro, C.: Time evolution of a Vlasov-Poisson plasma with magnetic confinement. Kinet. Release Models 5, 729-742 (2012) · Zbl 1263.82052 · doi:10.3934/krm.2012.5.729 |
| [4] | Caprino, S., Cavallaro, G., Marchioro, C.: On a magnetically confined plasma with infinite charge. SIAM J. Math. Anal. 46, 133-164 (2014) · Zbl 1296.82059 · doi:10.1137/130916527 |
| [5] | Caprino, S., Cavallaro, G., Marchioro, C.: Remark on a magnetically confined plasma with infinite charge. Rend. Mat. Appl. 35, 69-98 (2014) · Zbl 1315.82023 |
| [6] | Caprino, S., Cavallaro, G., Marchioro, C.: On a Vlasov-Poisson plasma confined in a torus by a magnetic mirror. J. Math. Anal. Appl. 427, 31-46 (2015) · Zbl 1339.82011 · doi:10.1016/j.jmaa.2015.02.012 |
| [7] | Caprino, S., Cavallaro, G., Marchioro, C.: Existence and uniqueness of the time evolution for a plasma with infinite charge in \[\mathbb{R}^3\] R3. Commun. Partial Differ. Equ. 40, 1-29 (2015) · Zbl 1320.82062 · doi:10.1080/03605302.2014.944267 |
| [8] | Caprino, S., Cavallaro, G., Marchioro, C.: A Vlasov-Poisson plasma with unbounded mass and velocities confined in a cylinder by a magnetic mirror. Kinet. Relat. Models 9, 657-687 (2016) · Zbl 1364.82061 · doi:10.3934/krm.2016011 |
| [9] | Caprino, S., Cavallaro, G., Marchioro, C.: The Vlasov-Poisson equation in \[\mathbb{R}^3\] R3 with infinite charge and velocities. arxiv: 1608.02336 · Zbl 1320.82062 |
| [10] | Castella, F.: Propagation of space moments in the Vlasov-Poisson equation and further results. Ann. Inst. Henri Poincaré 16, 503-533 (1999) · Zbl 1011.35034 · doi:10.1016/S0294-1449(99)80026-2 |
| [11] | Chen, Z.: The Vlasov-Poisson system with low steady asymptotics. J. Math. Phys. 56, 121503 (2015) · Zbl 1333.35289 · doi:10.1063/1.4938079 |
| [12] | Chen, Z., Zhang, X.: Global existence to the Vlasov-Poisson system and propagation of moments without assumption of finite kinetic energy. Commun. Math. Phys. 343, 851-879 (2016) · Zbl 1339.35317 · doi:10.1007/s00220-016-2616-9 |
| [13] | Desvillettes, L., Miot, E., Saffirio, C.: Polynomial propagation of moments and global existence for a Vlasov-Poisson system with a point charge. Ann. Inst. H. Poincaré Anal. Non Linéaire 32, 373-400 (2015) · Zbl 1323.35178 · doi:10.1016/j.anihpc.2014.01.001 |
| [14] | Gasser, I., Jabin, P.-E., Perthame, B.: Regularity and propagation of moments in some nonlinear Vlasov systems. Proc. R. Soc. Edinb. Sect. A 130, 1259-1273 (2000) · Zbl 0984.35102 · doi:10.1017/S0308210500000676 |
| [15] | Glassey, R.: The Cauchy Problem in Kinetic Theory. SIAM, Philadelphia, PA (1996) · Zbl 0858.76001 · doi:10.1137/1.9781611971477 |
| [16] | Jabin, P.-E.: The Vlasov-Poisson system with infinite mass and energy. J. Stat. Phys. 103, 1107-1123 (2001) · Zbl 1126.82327 · doi:10.1023/A:1010321308267 |
| [17] | Lions, P.-L., Perthame, B.: Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system. Invent. Math. 105, 415-430 (1991) · Zbl 0741.35061 · doi:10.1007/BF01232273 |
| [18] | Loeper, G.: Uniqueness of the solution to the Vlasov-Poisson system with bounded density. J. Math. Pures Appl. 9(86), 68-79 (2006) · Zbl 1111.35045 · doi:10.1016/j.matpur.2006.01.005 |
| [19] | Marchioro, C., Miot, E., Pulvirenti, M.: The Cauchy problem for the 3-D Vlasov-Poisson system with point charges. Arch. Ration. Mech. Anal. 201, 1-26 (2011) · Zbl 1321.76081 · doi:10.1007/s00205-010-0388-5 |
| [20] | Pallard, C.: Moment propagation for weak solutions to the Vlasov-Poisson system. Commun. Partial Differ. Equ. 37, 1273-1285 (2012) · Zbl 1387.76115 · doi:10.1080/03605302.2011.606863 |
| [21] | Pallard, C.: Space moments of the Vlasov-Poisson system: propagation and regularity. SIAM J. Math. Anal. 46, 1754-1770 (2014) · Zbl 1298.76241 · doi:10.1137/120881178 |
| [22] | Pankavich, S.: Global existence for the three dimensional Vlasov-Poisson system with steady spatial asymptotics. Commun. Partial Differ. Equ. 31, 349-370 (2006) · Zbl 1100.35066 · doi:10.1080/03605300500358004 |
| [23] | Pfaffelmoser, K.: Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data. J. Differ. Equ. 95, 281-303 (1992) · Zbl 0810.35089 · doi:10.1016/0022-0396(92)90033-J |
| [24] | Salort, D.: Transport equations with unbounded force fields and application to the Vlasov-Poisson equation. Math. Model Method Appl. Sci. 19, 199-228 (2009) · Zbl 1165.82011 · doi:10.1142/S0218202509003401 |
| [25] | Schaeffer, J.: Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions. Commun. Partial Differ. Equ. 16, 1313-1335 (1991) · Zbl 0746.35050 · doi:10.1080/03605309108820801 |
| [26] | Schaeffer, J.: The Vlasov-Poisson system with steady spatial asymptotics. Commun. Partial Differ. Equ. 28, 1057-1084 (2003) · Zbl 1103.82314 · doi:10.1081/PDE-120021186 |
| [27] | Schaeffer, J.: Steady spatial asymptotics for the Vlasov-Poisson system. Math. Methods Appl. Sci. 26, 273-296 (2003) · Zbl 1030.35116 · doi:10.1002/mma.354 |
| [28] | Schaeffer, J.: Global existence for the Vlasov-Poisson system with steady spatial asymptotic behavior. Kinet. Relat. Models 5, 129-153 (2012) · Zbl 1250.35166 · doi:10.3934/krm.2012.5.129 |
| [29] | Wollman, S.: Global in time solution to the three-dimensional Vlasov-Poisson system. J. Math. Anal. Appl. 176, 76-91 (1993) · Zbl 0814.35105 · doi:10.1006/jmaa.1993.1200 |
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