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Consider the Landau equation with Coulomb potential in a periodic box. Wedevelop a new $L^{2}\rightarrow L^{\infty }$ framework to construct globalunique solutions near Maxwellian with small $L^{\infty }\ $norm. The first stepis to establish global $L^{2}$ estimates with strong velocity weight and timedecay, under the assumption of $L^{\infty }$ bound, which is further controlledby such $L^{2}$ estimates via De Giorgi's 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 onHolder estimates via De Giorgi's method \cite{golse2016harnack},\cite{golse2015holder}, and \cite{mouhot2015holder}.

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