  This paper proposes four fundamental requirements for establishing PDEs
(partial differential equations) modeling irreversible processes. We show that
the PDEs derived via the CDF (conservation-dissipation formalism) meet all the
requirements. In doing so, we find useful constraints on the freedoms of CDF
and point out that a shortcoming of the formalism can be remedied with the help
of the Maxwell iteration. It is proved that the iteration preserves the
gradient structure and strong dissipativeness of the CDF-based PDEs. A refined
formulation of the second law of thermodynamics is given to characterize the
strong dissipativeness, while the gradient structure corresponds to nonlinear
Onsager relations. Further advantages and limitations of CDF will also be
presented.
