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A weak-strong uniqueness principle says that a weak solution with the same data as a sufficiently regular classical solution agrees with that solution throughout its interval of regular existence, within the specified weak class. The conclusion is class-dependent: all bounded weak solutions of a prescribed $C^1$ linear transport field satisfy it, while nonlinear conservation laws require an admissibility condition such as an entropy solution condition.

For a complete $C^1$ velocity flow, the bounded weak solution of $u_t+F\cdot \nabla u=0$ is its initial value pulled back by the inverse flow. Its weak identity is $\int u({\phi }_t+\mathrm{div}(F\phi ))+\int u_0\phi (0)=0$; the divergence term is essential. Changing to flow coordinates gives $\int w{\partial }_t(J\zeta )+\int u_0\zeta (0)=0$. Approximation in the spatial parameter permits $\zeta =\eta /J$, since $J$ and $J_t$ are continuous and $J>0$ on compact sets. Hence ${\partial }_{t}w=0$ distributionally, proving uniqueness without requiring second derivatives of $F$.