Duan, Renjun; Zhang, Mei; Zhu, Changjiang \(L^{1}\) stability for the Vlasov-Poisson-Boltzmann system around vacuum. (English) Zbl 1096.76050 Math. Models Methods Appl. Sci. 16, No. 9, 1505-1526 (2006). Summary: Based on the global existence theory of Vlasov-Poisson-Boltzmann system around vacuum in \(N\)-dimensional phase space, we prove the uniform \(L^{1}\) stability of classical solutions for small initial data when \(N \geq 4\). In particular, we show that the stability can be established directly for the soft potentials, while for the hard potentials and hard sphere model it is obtained through the construction of some nonlinear functionals. These functionals thus generalize those constructed by S.-Y. Ha [J. Differ. Equations 215, No. 1, 178–205 (2005; Zbl 1069.76048)] for the case without force to capture the effect of the force term on the time evolution of solutions. In addition, the local-in-time \(L^{1}\) stability is also obtained for the case of \(N = 3\). Cited in 7 Documents MSC: 76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics 76X05 Ionized gas flow in electromagnetic fields; plasmic flow 45K05 Integro-partial differential equations Keywords:soft potentials; hard potentials Citations:Zbl 1069.76048 PDFBibTeX XMLCite \textit{R. Duan} et al., Math. Models Methods Appl. Sci. 16, No. 9, 1505--1526 (2006; Zbl 1096.76050) Full Text: DOI References: [1] DOI: 10.1007/BF00251506 · Zbl 0654.76074 [2] Bardos C., AIHP, Anal. Nonlinear. 2 pp 101– [3] DOI: 10.1080/00411458908214500 · Zbl 0699.35237 [4] Bellomo N., Mathematical Topics in Nonlinear Kinetic Theory (1988) · Zbl 0702.76005 [5] DOI: 10.1007/s002050050165 · Zbl 0938.35093 [6] DOI: 10.1142/S0218202501001513 · Zbl 1013.35011 [7] DOI: 10.1007/978-1-4612-1039-9 [8] DOI: 10.1007/978-1-4419-8524-8 [9] DOI: 10.1137/040621338 · Zbl 1111.82053 [10] DOI: 10.1080/03605309108820765 · Zbl 0737.35127 [11] DOI: 10.1016/j.jde.2006.01.010 · Zbl 1101.76051 [12] Duan R. J., Discr. Cont. Dyn. Syst. 16 pp 253– [13] Frénod E., Math. Mod. Meth. Appl. Sci. 10 pp 539– [14] DOI: 10.1137/1.9781611971477 [15] DOI: 10.1142/S0218202503002386 · Zbl 1049.81022 [16] DOI: 10.1002/cpa.10040 · Zbl 1027.82035 [17] DOI: 10.1007/s002200100391 · Zbl 0981.35057 [18] Ha S.-Y., Arch. Rational Mech. Anal. 171 pp 279– [19] DOI: 10.1512/iumj.2005.54.2555 · Zbl 1125.35060 [20] DOI: 10.1016/j.jde.2004.07.022 · Zbl 1069.76048 [21] DOI: 10.1002/(SICI)1097-0312(199912)52:12<1553::AID-CPA3>3.0.CO;2-S · Zbl 1034.35073 [22] DOI: 10.1007/s002200050787 · Zbl 0983.45007 [23] DOI: 10.1007/BF01026493 · Zbl 1084.82545 [24] Poupaud F., Math. Mod. Meth. Appl. Sci. 10 pp 1027– [25] Toscani G., Rend. Circ. Mat. Palermo(2) Suppl. 8 pp 419– [26] DOI: 10.1016/S1874-5792(02)80004-0 [27] DOI: 10.1016/j.jcp.2004.07.017 · Zbl 1067.82057 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.