On the $L^2$-moment closure of transport equations: The Cattaneo approximation

We consider the moment-closure approach to transport equations which arise in Mathematical Biology. We show that the negative $L^2$-norm is an entropy in the sense of thermodynamics, and it satisfies an $H$-theorem. With an $L^2$-norm minimization procedure we formally close the moment hierarchy for the first two moments. The closure leads to semilinear Cattaneo systems, which are closely related to damped wave equations. In the linear case we derive estimates for the accuracy of this moment approximation. The method is used to study reaction-transport models and transport models for chemosensitive movement. With this method also order one perturbations of the turning kernel can be treated - in extension of an earlier theory on the parabolic limit of transport equations (Hillen and Othmer 2000). Moreover, this closure procedure allows us to derive appropriate boundary conditions for the Cattaneo approximation. Finally, we illustrate that the Cattaneo system is the gradient flow of a weighted Dirichlet integral and we show simulations.
The moment closure for higher order moments and for general transport models will be studied in a second paper.