Moment Propagation for the Vlasov--Poisson System with Gyroscopic Forces, Velocity Diffusion, and Specular Reflection

This paper studies the three-dimensional Vlasov–Poisson system with gyroscopic forces and velocity diffusion. For the Cauchy problem in ${\mathbb{R}}^3$ and the specular–Dirichlet problem in bounded domains, we prove the propagation of velocity moments of order $k>2$ for weak solutions with bounded $L^1\cap L^{\mathrm{\infty }}$ norms and kinetic energy. The proof applies Serre’s compensated integrability to weighted mass–momentum tensors to derive density estimates. A finite iteration alternating moment propagation with improved density integrability then closes the estimates. These estimates provide a simplified approach to moment propagation and yield global continuation of classical solutions.