The 3D Vlasov–Poisson–Landau System Near 1D Local Maxwellians

Abstract

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.