 Solution

If $H=a(v)$, the characteristics have constant velocity variable $V(t)=\eta$ and $X(t)=\xi +t\nabla a(\eta )$. This flow is globally defined for every finite $t$ at each $(\xi ,\eta )$. ThereforeDirect differentiation gives $f_t+\nabla a(v)\cdot {\nabla }_{x}f=0$ and the initial value at $t=0$.