 Solution

For every test function $\varphi \in C_c^{\infty }({\mathbb{R}}_t\times {\mathbb{R}}_x^3\times {\mathbb{R}}_v^3)$, the distributional weak solution satisfiesThere 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 ${\partial }_{t}f+v\cdot {\nabla }_{x}f$. 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.