An $L^2$ to $L^{\mathrm{\infty }}$ Framework for the Landau Equation

Consider the Landau equation with Coulomb potential in a periodic box. We develop a new $L^2\text{to}{L}^{\mathrm{\infty }}$ framework to construct global unique solutions near Maxwellian with small $L^{\mathrm{\infty }}$ norm. The first step is to establish global $L^2$ estimates with strong velocity weight and time decay, under the assumption of $L^{\mathrm{\infty }}$ 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)).