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The 3D Vlasov-Poisson-Landau system near 1D local Maxwellians. (English) Zbl 1462.35396

Summary: We consider the Cauchy problem on the Vlasov-Poisson-Landau system with the Coulomb interaction in the three dimensional space domain \(\mathbb{R} \times \mathbb{T}^2\). Although there have been extensive studies on global existence and large time behavior of solutions near global Maxwellians either in \(\mathbb{T}^3\) or \(\mathbb{R}^3\), it is unknown whether solutions can be constructed around some non-trivial asymptotic profiles. In this paper, we obtain the global solutions near a spatially one-dimensional local Maxwellian connecting two distinct global Maxwellians at \(x_1=\pm \infty\), where the fluid components of the local Maxwellian are the smooth approximate rarefaction wave of the corresponding full compressible Euler system in \(x_1\in \mathbb{R}\). We also prove the large time asymptotics of global solutions towards such planar rarefaction waves, and establish the propagation of the high-order moments and regularity of solutions with large amplitude. As a byproduct, all these results can be carried over to the pure Landau equation in the same setting.

MSC:

35Q83 Vlasov equations
35Q31 Euler equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
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