Conservation of Energy, Entropy Identity, and Local Stability for the Spatially Homogeneous Boltzmann Equation

Abstract

For nonsoft potential collision kernels with angular cutoff, we prove that under the initial condition f ₀(v)(1+|v|²+|logf ₀(v)|)∈L ¹(R ³), the classical formal entropy identity holds for all nonnegative solutions of the spatially homogeneous Boltzmann equation in the class L ^∞([0, ∞); L ¹ ₂(R ³))∩C ¹([0, ∞); L ¹(R ³)) [where L ¹ _s (R ³)={f∣f(v)(1+|v|²)^s/2∈L ¹(R ³)}], and in 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 ¹ _loc([0, ∞); L ¹ _2+β (R ³)) is also given.