An entropy chain rule on bounded Lipschitz domains

We prove an exact entropy chain rule for nonnegative densities satisfying a weak continuity equation with no boundary flux on any bounded Lipschitz domain. Finite kinetic action and Fisher information, together with finite initial entropy and weak continuity in time, imply absolute continuity of the entropy on the closed time interval. The formula holds for the prescribed flux, with vacuum allowed and without upper or lower density bounds. This extends the convex-domain chain rule to nonsmooth nonconvex domains. The proof reflects both the density and the flux across local boundary graphs. A truncated entropy identity then permits the localization terms to cancel before the logarithmic truncation is removed.