A L² to L^∞ approach for the Landau Equation

2016/10/17 by Jin‐Oh Kim, Yan Guo, Kim, Jinoh +3
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Gas Dynamics and Kinetic Theory

paper · pdf · doi:10.48550/arxiv.1610.05346

openalex publication_date 2016/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider the Landau equation with Coulomb potential in a periodic box. We develop a new L²→ L^∞ framework to construct global unique solutions near Maxwellian with small L^∞ norm. The first step is to establish global L² estimates with strong velocity weight and time decay, under the assumption of L^∞ bound, which is further controlled by such L² estimates via De Giorgi's method \citegolse2016harnack and \citemouhot2015holder. The second step is to employ estimates in S_p spaces to control velocity derivatives to ensure uniqueness, which is based on Holder estimates via De Giorgi's method \citegolse2016harnack, \citegolse2015holder, and \citemouhot2015holder.

Citations

• Hölder continuity of solutions to hypoelliptic equations with bounded measurable coefficients
• Entropy dissipation estimates for the Landau equation in the Coulomb\n case and applications

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