On the $L^2$-moment closure of transport equations: The general case

Transport equations are intensively used in Mathematical Biology. In this article the moment closure for transport equations for an arbitrary finite number of moments is presented. With use of a variational principle the closure can be obtained by minimizing the $L^2(V)$-norm with constraints. An $H$-Theorem for the negative $L^2$-norm is shown and the existence of Lagrange multipliers is proven. The Cattaneo closure is a special case for two moments and was studied in Part I (Hillen 2003). Here the general theory is given and the three moment closure for two space dimensions is calculated explicitly. It turns out that the steady states of the two and three moment systems are determined by the steady states of a corresponding diffusion problem.