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It seems as though equation 2 in page 15 should logically come before the preceding equation. The change in the volume of a fluid element is given by $ \frac{1}{V} \frac{dV}{dt} = \nabla \cdot \mathbf{U} $. The energy and volume are related in adiabatic expansion by

$ U \cdot V^{\gamma-1} = const \Rightarrow \frac{d}{dt} \ln E = -\left( \gamma - 1 \right) \frac{d}{dt} \ln V = - \left( \gamma -1 \right) \nabla \cdot \mathbf{U} $

For classical monatomic gasses, the relation between the energy and momentum is $ E \propto p^2 $ and $ \gamma = \frac{5}{3} $ so

$ \frac{d}{dt} \ln p = - \frac{1}{3} \nabla \cdot \mathbf{U} $

Incidently, in the ultra - relativistic case where $ E \propto p $ and $ \gamma = \frac{4}{3} $ we get the same result.