= Weak conservation laws depend on conserved variables
Two smooth differential systems related by a nonlinear change of dependent variables need not define the same <weak solutions>. If $U_t+F(U)_x=0$, a smooth function $V=g(U)$ satisfies $V_t+g'(U)F'(U)U_x=0$ for smooth $U$; even when this can be written as another divergence, its <Rankine-Hugoniot conditions> need not be equivalent to those in $U$. The smooth chain rule does not justify multiplying jump distributions by discontinuous coefficients.
For example the <shallow water equations> can be written smoothly as $h_t+(hv)_x=0$ and $v_t+(h+v^2/2)_x=0$. Mechanical mass and <momentum conservation> instead select $h_t+(hv)_x=0$ and $(hv)_t+(hv^2+h^2/2)_x=0$. These systems agree for smooth positive $h$, but their jump laws differ. The states $(h,v)=(2,\sqrt3/2)$ and $(1,0)$ satisfy the mass/momentum jump laws with speed $\sqrt3$, while the velocity-form jump equation would demand $11/(4\sqrt3)$. A model for discontinuities must specify the conservative variables or an equivalent admissible regularization, rather than using smooth equivalence alone.
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