Boltzmann Equation with a Large Potential in a Periodic Box

The stability of the Maxwellian of the Boltzmann equation with a large amplitude external potential $Φ$ has been an important open problem. In this paper, we resolve this problem with a large $C3-$potential in a periodic box $\mathbb{T}^d$, $d \geq 3$. We use [1] in $L^p-L^{\infty}$ framework to establish the well-posedness and the $L^{\infty}-$stability of the Maxwellian $μ_E(x,v)=\exp\{-\frac{|v|^2}{2}-Φ(x)\}$.

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