Distributional solutions of the Boltzmann equation with infinite energy

We prove an existence theorem of distributional solutions to the Boltzmann equation for soft potentials with angular cut-off and initial data close to the local Maxwellian M0=exp(−|x−ξ|22). The solutions constructed in this paper have infinite mass and energy since they are located in a neighborhood of the Maxwellian M(t,x,ξ)=exp(−|x−ξ(1+t)|22) (t⩾0t⩾0).