Balance laws and centro velocity in dissipative systems

Starting with a density that is conserved for a dynamical system when dissipation is ignored, a local conservation law is derived for which the total flux (integrated over the spatial domain) is unique. When dissipation is incorporated, the conservation law becomes a balance law. The contribution due to dissipation in this balance law is split in a unique way in a part that is proportional to the density and in a divergence expression that adds to the original (conservative) flux density; the total additional flux is uniquely defined. It is shown that these total fluxes appear in the expression for the centro velocity, i.e., in the velocity of the center of gravity of the density, which shows that this velocity can be defined in a unique way (in contrast to a local velocity). Applications to the Korteweg–de Vries–Burgers equations and to the incompressible Navier–Stokes equations are given