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

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Abstract

We consider the Vlasov–Poisson–Landau system, a classical model for a dilute collisional plasma interacting through Coulombic collisions and with its self-consistent electrostatic field. We establish global stability and well-posedness near the Maxwellian equilibrium state with decay in time and some regularity results for small initial perturbations, in any general bounded domain (including a torus as in a tokamak device), in the presence of specular reflection boundary condition. We provide a new improved \(L^{2}\rightarrow L^{\infty }\) framework: \(L^{2}\) energy estimate combines only with \(S_{ }^{p}\) estimate for the ultra-parabolic equation.

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Notes

  1. The constant \(c_{12}=2\pi e_1^2 e_2^2 \ln \Lambda \), \(\ln \Lambda =\ln \big (\frac{\lambda _D}{b_0}\big )\), where \(\lambda _D=\big (\frac{T}{4\pi n_e e^2}\big )^{\!1/2}\) is the Debye shielding distance and \(b_0=\frac{e^2}{3T}\)  a typical “distance of closest approach” for a thermal particle.

  2. The authors used spherical-type coordinates to make the map almost globally defined; here we just prefer the standard coordinates for simplicity.

  3. We use the column vector convention in the following matrix operation expressions.

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Acknowledgements

The authors would like to thank Dr. Sona Akopian for many helpful discussions. They are also grateful to Dr. Timur Yastrzhembskiy for careful reading and pointing out a few important issues in an earlier version of the manuscript. Yan Guo’s research is supported in part by NSF Grant DMS-2106650. Hongjie Dong is partially supported by the Simons Foundation, Grant No. 709545, a Simons fellowship, Grant No. 007638, and the NSF under Agreement DMS-2055244.

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Authors and Affiliations

  1. Brown University, Providence, USA

    Hongjie Dong & Yan Guo

  2. Institute for Pure and Applied Mathematics, University of California, Los Angeles, USA

    Zhimeng Ouyang

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  1. Hongjie Dong
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  2. Yan Guo
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  3. Zhimeng Ouyang
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Correspondence to Yan Guo.

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Communicated by T.-P. Liu.

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Dong, H., Guo, Y. & Ouyang, Z. The Vlasov–Poisson–Landau System with the Specular-Reflection Boundary Condition. Arch Rational Mech Anal 246, 333–396 (2022). https://doi.org/10.1007/s00205-022-01818-9

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