WEAK SOLUTIONS OF THE BOLTZMANN-EQUATION AND ENERGY-CONSERVATION

An inequality which guarantees that both the gain and the loss term in the collision integral of the Boltzmann equation are in L(1) under a truncation which only depends upon the deflection angle and the relative speed and thus amounts to an acceptable assumption on the cross section was recently proved by the author. This inequality, which refers to solutions depending on just one space variable, is generalized here to show that one can dispense with the concept of renormalized solution used in the existence proof of DiPerna and Lions. In addition, new inequalities, including one related to energy conservation, are proved.