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 M 0 = exp ( − | x − ξ | 2 2 ) . 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 ) | 2 2 ) ( t ⩾ 0 ).