Dissipation

An equation is informally called dissipative if the high frequency components of the solution decay in energy as one evolves forward in time. The model instance of a dissipative equation is the heat equation

$u_{t}=\Delta u.$

Dissipation should be compared with dispersion, in which the high frequencies move around but do not actually decrease in energy. The two are somewhat similar but dsitinct. For instance, dispersive equations tend to be time-reversible, whereas dissipative equations are not. While dispersive equations are often Hamiltonian and are the Euler-Lagrange equation for some variational problem, dissipative equations are usually non-Hamiltonian and are generated by a gradient flow for a variational problem rather than by locating a critical point.