The Vlasov–Poisson–Landau System with the Specular-Reflection Boundary Condition

Dong, Hongjie; Guo, Yan; Ouyang, Zhimeng

doi:10.1007/s00205-022-01818-9

Cited by 7 publications

(3 citation statements)

References 28 publications

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“…Nevertheless, with the aid of the Velocity Lemmas and tools from differential geometry, classical solutions to the Vlasov-Poisson system have been obtained by the Eulerian method or the Lagrangian method under suitable conditions on the initial datum near the grazing set, as shown in [24,29,[31][32][33] for half space or convex domain and Neumann boundary condition or Dirichlet boundary condition. Recently, based on the L 2 − L ∞ framework developed by Guo [26], regularity of the solutions to the initialboundary value problems of the kinetic equations has been established in [5,18,27,28] and so on.…”

Section: Background and The Contribution Of The Papermentioning

confidence: 99%

The plasma-charge model in a convex domain

2024

Nonlinearity

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The aim of this paper is to study the initial-boundary value problems of a Vlasov type system in a convex domain, so called the plasma-charge model, in which there are two kinds of singular sets, one caused by the boundary effect, the other by the heavy point charges. We prove the local existence of classical solutions for the case that the point charges are moving and global existence of classical solutions for the case that the point charges are fixed away from the boundary. The crucial tools are the extended Velocity Lemma for the plasma-charge model and the Pfaffelmoser’s method developed by Hwang and Velázquez (2010 Arch. Ration. Mech. Anal. 195 763–96) and Marchioro et al (2011 Arch. Ration. Mech. Anal. 201 1–26). In the Pfaffelmoser’s argument, a new idea is that the plasma particles can only be close to one of the singular sets during the time interval [ t − δ , t ] with small length δ, which allows us to obtain the global existence for the fixed point charges case by adapting the techniques established by Hwang et al (2013 Discrete Contin. Dyn. Syst. 33 723–37; 2010 Arch. Ration. Mech. Anal. 195 763–96) and Marchioro et al (2011 Arch. Ration. Mech. Anal. 201 1–26) to the corresponding singular sets respectively.

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Section: Background and The Contribution Of The Papermentioning

confidence: 99%

The plasma-charge model in a convex domain

2024

Nonlinearity

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“…Duan-Liu-Sakamoto-Strain [11] proved the global existence for the Landau and non-cutoff Boltzmann equation in a finite channel. Dong-Guo-Ouyang [8] established the global existence for VPL system in general bounded domain with the specular boundary condition.…”

Section: Spatial Domain and Boundary Conditionmentioning

confidence: 99%

Low regularity solutions for the Vlasov–Poisson–Landau/Boltzmann system

Deng

Duan

2023

Nonlinearity

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In the paper, we are concerned with the nonlinear Cauchy problem on the Vlasov–Poisson–Landau/Boltzmann system around global Maxwellians in a torus or a finite channel. The main goal is to establish the global existence and large time behaviour of small amplitude solutions for a class of low regularity initial data. The molecular interaction type is restricted to the case of hard potentials for two classical collision operators because of the effect of the self-consistent forces. The result extends the one by Duan–Liu–Sakamoto-Strain (Duan et al 2021 Commun. Pure Appl. Math. 74 932–1020) for the pure Landau/Boltzmann equation to the case of the VPL and VPB systems.

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“…The global classical solutions of the VPL system around local Maxwellians with rarefaction waves either in the one-dimensional line with slab symmetry or in the three-dimensional infinite channel domain was obtained by Duan-Yu [25,29]. In addition to those works, great contributions also have been done in many other kinds of topics of the VPL system, for instance [7,18,20,30] and the references therein.…”

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confidence: 99%

The Vlasov-Poisson-Landau system near a local Maxwellian

Duan

2020

Advances in Mathematics

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The Vlasov–Poisson–Landau System with the Specular-Reflection Boundary Condition

Cited by 7 publications

References 28 publications

The plasma-charge model in a convex domain

The plasma-charge model in a convex domain

Low regularity solutions for the Vlasov–Poisson–Landau/Boltzmann system

The Vlasov-Poisson-Landau system near a local Maxwellian

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