  We provide a thorough study of stability of the 1-D continuity equation,
which models many physical conservation laws. In our system-theoretic
perspective, the velocity is considered to be an input. An additional input
appears in the boundary condition (boundary disturbance). Stability estimates
are provided in all Lp state norms with p>1, as well as in the sup norm.
However, in our Input-to-State Stability estimates, the gain and overshoot
coefficients depend on the velocity. Moreover, the logarithmic norm of the
state appears instead of the usual norm. The obtained results can be used in
the stability analysis of larger models that contain the continuity equation.
In particular, it is shown that the obtained results can be used in a
straightforward way for the stability analysis of non-local, nonlinear
manufacturing models under feedback control.
