Conservation of energy, entropy identity, and local stability for the spatially homogeneous Boltzmann equation

For nonsoft potential collision kernels with angular cutoff, we prove that under the initial condition f(0)(v)(1 + \v\(2) + \log f(0)(v)\) epsilon L-1(R-3), the classical formal entropy identity holds for all nonnegative solutions of the spatially homogeneous Boltzmann equation in the class L-infinity([0, infinity); L-2(1)(R-3)) boolean AND C-1([0, infinity); L-1(R-3)) [where L-s(1)(R-3)= {f \ f(v)(1 + \v\(2))(s/2) epsilon L-1 (R-3)}] , this class, the nonincrease of energy always implies the conservation of energy and therefore the solutions obtained all conserve energy. Moreover, for hard potentials and the hard-sphere model, a local stability result for conservative solutions (i.e., satisfying the conservation of mass, momentum, and energy) is obtained. As an application of the local stability, a sufficient and necessary condition on the initial data f(0) such that the conservative solutions f belong to L-loc(1)(0, infinity); L-2 + beta(1)(R-3)) is also given.