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

摘要

For nonsoft potential collision kernels with angular cutoff, we prove that under the initial condition f_0(#upsilon#) (1+|#upsilon#|~2+|log f_0(#upsilon#)|) implied by 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)) intersect C~1 ([0, propor. to]); L~1(R~3)) [where L_s~1(R~3)={f|f(#epsilon#)(1+|#upsilon#|~2)~(s/2) implied 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_(loc)~1([0, propor. to]; L_(2+#beta#)~1(R~3)) is also given.