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|^α + |x-v|²) \, f₀ ∈ L¹ instead of (|x|² + |v|²) \, f₀ ∈ L¹, 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.