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 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 on Holder estimates via De Giorgi's method \cite{ golse2016harnack}, \cite{ golse2015holder}, and \cite{ mouhot2015holder}.
Categories: math.AP