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

Abstract

Consider the Landau equation with Coulomb potential in a periodic box. We develop a new $L^2\to L^{\infty }$ 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}.

References in corpus (2)

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