A $L^{2}$ to $L^{infty}$ approach for the Landau Equation

Consider the Landau equation with Coulomb potential in a periodic box. We develop a new $L^{2}rightarrow L^{infty }$ 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 Giorgis method cite{golse2016harnack} and cite{mouhot2015holder}. 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 Giorgis method cite{golse2016harnack}, cite{golse2015holder}, and cite{mouhot2015holder}.