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The flux Jacobian is the Jacobian matrix of the flux in a system of conservation laws. Its eigenvalues are the local characteristic speeds, and its left and right eigenvectors select the corresponding wave variables and directions in state space.

Local mass conservation of each particle species gives the system of conservation lawsFor $\mathbf{\varphi }=({\varphi }_1,{\varphi }_2{)}^T$ and flux $F_i={\varphi }_i{W}_i$, the flux Jacobian isWriting its entries as $A_{ij}$, the two characteristic speeds are its eigenvaluesThe discriminant is nonnegative because $A_{12}{A}_{21}={\varphi }_1{\varphi }_2{\hat{W}}_1{\hat{W}}_2\ge 0$, so positive concentrations of two settling species give real characteristics.

The Rankine-Hugoniot condition expresses conservation laws across a moving discontinuity of a weak solution. Across a discontinuity $x=\sigma t$ in $u_t+f(u{)}_x=0$, with left and right states $u_{-}$ and $u_{+}$, conservation requiresWriting $[u]=u_{+}-u_{-}$ and $[f(u)]=f(u_{+})-f(u_{-})$ gives $\sigma [u]=[f(u)]$. The same relation holds componentwise for a system of conservation laws; in compressible flow, conservation of mass, momentum and energy supplies the jump conditions for a shock wave. Integrating the weak solution identity over a thin space-time region crossing the discontinuity yields this balance between the swept-up conserved quantity and the difference of fluxes.

A Riemann problem for a one-dimensional system of conservation laws ${\partial }_{t}U+{\partial }_{x}f(U)=0$ has two constant initial states, $U_L$ for $x<0$ and $U_R$ for $x>0$. The data and equation are invariant under $(x,t)\mapsto (\lambda x,\lambda t)$. When the physically admissible entropy solution is unique, this invariance makes it a self-similar solution depending on $x/t$. For ideal-gas compressible flow, the characteristic speeds are $u-c,u,u+c$. The two acoustic families carry shock waves or rarefaction waves, while the middle family carries a contact discontinuity. Sufficiently strong separating velocities can instead create a vacuum region.