An \(L^2\) to \(L^\infty \) Framework for the Landau Equation

Abstract

Consider the Landau equation with Coulomb potential in a periodic box. We develop a new \(L^{2}\ \text{to}\ L^{\infty }\) framework to construct global unique solutions near Maxwellian with small \(L^{\infty }\) norm. The first step is to establish global \(L^{2}\) estimates with strong velocity weight and time decay, under the assumption of \(L^{\infty }\) bound, which is further controlled by such \(L^{2}\) estimates via De Giorgi’s method (Golse et al. in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19(1), 253–295 (2019), Imbert and Mouhot in arXiv:1505.04608 (2015)). The second step is to employ estimates in \(S_{p}\) spaces to control velocity derivatives to ensure uniqueness, which is based on Hölder estimates via De Giorgi’s method (Golse et al. in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19(1), 253–295 (2019), Golse and Vasseur in arXiv:1506.01908 (2015), Imbert and Mouhot in arXiv:1505.04608 (2015)).

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Acknowledgements

Yan Guo is supported in part by NSF grant #DMS-1810868, Chinese NSF Grant #10828103, as well as a Simon Fellowship. Hyung Ju Hwang was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF-2017R1E1A1A03070105, NRF-2019R1A5A1028324).

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Correspondence to Yan Guo or Hyung Ju Hwang.

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Kim, J., Guo, Y. & Hwang, H.J. An \(L^2\) to \(L^\infty \) Framework for the Landau Equation. Peking Math J 3, 131–202 (2020). https://doi.org/10.1007/s42543-019-00018-x

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Keywords

  • Landau equation
  • Weak solution
  • Existence and uniqueness
  • L2 to \(L^\infty \) framework

Mathematics Subject Classification

  • 35Qxx