= Adjoint transport equation
The <adjoint transport equation> moves a <transport equation> operator from a solution to a <test function> by <integration by parts>. For $L u=\partial_tu+b\cdot\nabla u$, its formal space-time <adjoint operator> is
$$
L^*\varphi=-\partial_t\varphi-\operatorname{div}(b\varphi).
$$
Indeed $\int(Lu)\varphi=\int uL^*\varphi$ when both functions have compact <support> away from the time endpoints. The term $-(\operatorname{div}b)\varphi$ is essential for a compressible <velocity field>. By contrast, the conservative operator $M\rho=\partial_t\rho+\operatorname{div}(b\rho)$ has $M^*\varphi=-\partial_t\varphi-b\cdot\nabla\varphi$.
For a smooth complete <characteristic flow map> with compact <supports> remaining bounded on $[0,T]$, solve $L^*\varphi=g$ backward with $\varphi(T)=0$, for a smooth compactly supported $g$. <Integration by parts> in time gives $\int_0^T\int u g=\int u_0\varphi(0)$ for a <weak solution> of $Lu=0$. Thus zero <initial condition> forces zero pairing with every such $g$, proving uniqueness. The <characteristic curves> solve this terminal problem with a <divergence> weight; omitting that weight would test a different equation.
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