Conservation law in divergence form

A scalar conservation law with source has the form

\partial_tq+\nabla\cdot J=s,

where $q$ is a density, $J$ its spatial flux, and $s$ a source. It is homogeneous when $s=0$ . The spatial divergence gives a conservative or divergence form of the evolution equation.

Integral meaning

For a fixed bounded region $\Omega$ with smooth boundary, sufficient regularity and the divergence theorem give

\frac{d}{dt}\int_\Omega q\,dx =-\int_{\partial\Omega}J\cdot n\,dS+\int_\Omega s\,dx,

where $n$ is the outward unit normal. The negative sign accounts for outward flux reducing the amount inside.

Systems

For vector-valued densities, each component has its own flux. These fluxes form a matrix, and the system uses its row divergence.