= Solution
For every <test function> $\phi\in C_c^\infty(\mathbb R_t\times\mathbb R_x^3\times\mathbb R_v^3)$, the <distributional weak solution> satisfies
$$
\boxed{-\int f(t,x,v)\bigl(\partial_t\phi+v\cdot\nabla_x\phi\bigr)\,dt\,dx\,dv
=\int g(t,x,v)\phi(t,x,v)\,dt\,dx\,dv}.
$$
There is no initial-time boundary term because the <test functions> are on the whole time line. In <divergence> notation the flux in the seven coordinates is $(f,vf,0)$, whose <divergence> is $\partial_tf+v\cdot\nabla_xf$. For a general $N$-dimensional <divergence> the introductory flux must be vector-valued; the scalar codomain printed for $A$ is dimensionally inconsistent when $N>1$. The transport identity above has the correct flux dimensions.
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