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1. The Landau Equation with the Specular Reflection Boundary Condition.

   Guo, Yan; Hwang, Hyung Ju; Jang, Jin Woo; Ouyang, Zhimeng (January 2020, Archive for Rational Mechanics and Analysis)

   The existence and stability of the Landau equation (1936) in a general bounded domain with a physical boundary condition is a long-outstanding open problem. This work proves the global stability of the Landau equation with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. The highlight of this work also comes from the low-regularity assumptions made for the initial distribution. This work generalizes the recent global stability result for the Landau equation in a periodic box (Kim et al. in Peking Math J, 2020). Our methods consist of the generalization of the wellposedness theory for the Fokker–Planck equation (Hwang et al. SIAM J Math Anal 50(2):2194–2232, 2018; Hwang et al. Arch Ration Mech Anal 214(1):183–233, 2014) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi–Nash–Moser theory for the kinetic Fokker–Planck equations (Golse et al. Ann Sc Norm Super Pisa Cl Sci 19(1):253–295, 2019) and the Morrey estimates (Bramanti et al. J Math Anal Appl 200(2):332–354, 1996) to further control the velocity derivatives, which ensures the uniqueness. Our methods provide a new understanding of the grazing collisions in the Landau theory for an initial-boundary value problem. 

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2. Global existence for an isotropic modification of the Boltzmann equation

   https://doi.org/10.1016/j.jfa.2024.110423  

   Snelson, Stanley (June 2024, Journal of Functional Analysis)

   Motivated by the open problem of large-data global existence for the non-cutoff Boltzmann equation, we introduce a model equation that in some sense disregards the anisotropy of the Boltzmann collision kernel. We refer to this model equation as isotropic Boltzmann by analogy with the isotropic Landau equation introduced by Krieger and Strain (2012) [35]. The collision operator of our isotropic Boltzmann model converges to the isotropic Landau collision operator under a scaling limit that is analogous to the grazing collisions limit connecting (true) Boltzmann with (true) Landau. Our main result is global existence for the isotropic Boltzmann equation in the space homogeneous case, for certain parts of the “very soft potentials” regime in which global existence is unknown for the space homogeneous Boltzmann equation. The proof strategy is inspired by the work of Gualdani and Guillen (2022) [22] on isotropic Landau, and makes use of recent progress on weighted fractional Hardy inequalities. 

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3. Experimental verification of the Landau–Lifshitz equation

   https://doi.org/10.1088/1367-2630/ac1554  

   Nielsen, C F; Justesen, J B; Sørensen, A H; Uggerhøj, U I; Holtzapple, R (August 2021, New Journal of Physics)

   Abstract The Landau–Lifshitz (LL) equation has been proposed as the classical equation to describe the dynamics of a charged particle in a strong electromagnetic field when influenced by radiation reaction. Until recently, there has been no clear experimental verification. However, aligned crystals have remedied the situation: here, as in Nielsen et al CERN NA63 Collaboration (2020 Phys. Rev. D 102 052004), we report on a quantitative experimental test of the LL equation by measuring the emission spectra of electrons and positrons penetrating aligned single crystals. The recorded spectra are in remarkable agreement with simulations based on the LL equation of motion with moderate quantum corrections for recoil and, in the case of electrons in axially aligned crystals, spin and reduced radiation intensity. 

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4. Hilbert expansion for Coulomb collisional kinetic models

   https://doi.org/10.1090/qam/1689  

   Ouyang, Zhimeng; Wu, Lei; Xiao, Qinghua (March 2024, Quarterly of Applied Mathematics)

   The relativistic Vlasov-Maxwell-Landau (r-VML) system and the relativistic Landau (r-LAN) equation are fundamental models that describe the dynamics of an electron gas. In this paper, we introduce a novel weighted energy method and establish the validity of the Hilbert expansion for the one-species r-VML system and r-LAN equation. As the Knudsen number shrinks to zero, we rigorously demonstrate the relativistic Euler-Maxwell limit and relativistic Euler limit, respectively. This successfully resolves the long-standing open problem regarding the hydrodynamic limits of Landau-type equations. 

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5. Nonlinear steepest descent on a torus: a case study of the Landau–Lifshitz equation

   https://doi.org/10.1088/1361-6544/adbe22  

   Desiraju, Harini; Its, Alexander; Prokhorov, Andrei (March 2025, Nonlinearity)

   Abstract We obtain rigorous large time asymptotics for the Landau–Lifshitz (LL) equation in the soliton free case by extending the nonlinear steepest descent method to genus 1 surfaces. The methods presented in this paper pave the way to a rigorous analysis of other integrable equations on the torus and enable asymptotic analysis on different regimes of the LL equation. 

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