BOLTZMANN-EQUATION WITH INFINITE ENERGY - RENORMALIZED SOLUTIONS AND DISTRIBUTIONAL SOLUTIONS FOR SMALL INITIAL DATA AND INITIAL DATA CLOSETO A MAXWELLIAN

We prove new existence results for the Boltzmann equation with an init ial data with infinite energy, In the framework of renormalized soluti ons we assume (\x\(alpha) + \x - v\(2)) f(0) is an element of L-1 inst ead of (\x\(2) + \v\(2)) f(0) is an element of L-1, and we show new a priori estimates. In the framework of distributional solutions we trea t 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. H ence, our theory does not require that the initial data decrease in bo th variables x and v.