Errata: Weak solutions of the Boltzmann equation and energy conservation

Abstract

In two recent papers, the author discussed inequalities which guarantee that both the gain and the loss term in the collision integral of the Boltzmann equation are in L 1 under a suitable truncation. Due to an oversight, the truncation indicated in the above-mentioned papers is not correct. A correct truncation, discussed here, only depends upon the relative speed (and not upon the deflection angle) and amounts to an acceptable assumption on the cross section. The inequality on the gain and loss terms, which refers to solutions depending on just one space variable, then remains true and guarantees that one can dispense with the concept of renormalized solution used in the existence proof of DiPerna and Lions. The key to the result presented here is the inequality related to energy conservation, proved in the second of the previously mentioned papers.