Bounded solutions of the Boltzmann equation In the whole space

We construct bounded classical solutions of the Boltzmann equa-tion in the whole space without specifying any limit behaviors at the spatial infinity and without assuming the smallness condition on initial data. More precisely, we show that if the initial data is non-negative and belongs to a uniformly local Sobolev space in the space variable and a standard Sobolev space with Maxwellian type decay property in the velocity variable, then the Cauchy problem of the Boltzmann equation possesses a unique non-negative local solution in the same function space, both for the cutoff and non-cutoff collision cross section with mild singularity. The known solutions such as so-lutions on the torus (space periodic solutions) and in the vacuum (solutions vanishing at the spatial in_nity), and solutions in the whole space having a limit equilibrium state at the spatial infinnity are included in our category. © American Institute of Mathematical Sciences.