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 \(f_0(v)(1+|v|^2+ |\log f_0 (v) |)\in L^1(\mathbb{R}^3)\), the classical formal entropy identity holds for all nonnegative solutions of the spatially homogeneous Boltzmann equation in the class \(L^\infty([0,\infty);\;L^1_2(\mathbb{R}^2)\cap C^1([0,\infty)\); \(L^1(\mathbb{R}^3))\) [where \(L^1_s(\mathbb{R}^3)= \{f\mid f(v)(1+ |v|^2)^{s/2}\in L^1(\mathbb{R}^3) \}]\), 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 is such that the conservative solutions \(f\) belong to \(L^l_{\text{loc}} ([0,\infty);\;L^1_{2+\beta} (\mathbb{R}^3))\) is also given.
scientific article; zbMATH DE number 1404153