Boltzmann equation with infinite energy: renormalized solutions and distributional solutions for small initial data and initial data close to a Maxwellian

We prove new existence results for the Boltzmann equation with an initial data with infinite energy. In the framework of renormalized solutions we assume $(|x|^\alpha + |x-v|^2) \, f_0 \in L^1$ instead of $(|x|^2 + |v|^2) \, f_0 \in L^1$, and we show new a priori estimates. In the framework of distributional solutions we treat small initial data compared to a Maxwellian of the type $\exp ( - |x-v|^2/2)$. We also treat initial data close enough to such a Maxwellian. Hence, our theory does not require that the initial data decrease in both variables x and v.