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\(L^{1}\) stability for the Vlasov-Poisson-Boltzmann system around vacuum. (English) Zbl 1096.76050

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 \(N = 3\).

MSC:

76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
45K05 Integro-partial differential equations

Citations:

Zbl 1069.76048
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References:

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