Local well-posedness for the periodic Boltzmann equation with constant collision kernel

We study the Boltzmann equation with the constant collision kernel in the case of spatially periodic domain ${\mathbb{T}}^d$, $d\ge 2$. Using the existing techniques from nonlinear dispersive PDEs, we prove the local well-posedness result in $L_v^{2,r}{H}_x^s$ for $s>\frac{d}{2}-\frac{1}{4}$ and $r>\frac{d}{2}$. To reach the result, the main tool we establish is the $L^4$ Strichartz estimate for solutions to the corresponding linear equation.