Thermodynamic Structure and Composition in Nonlinear Convection-Diffusion

Nonlinear convection--diffusion systems play a central role in transport phenomena, including mass transfer, heat transfer, porous-media transport, and coupled continuum processes with source, exchange, and interface effects. In such systems, the key question is often not only which governing partial differential equation is used, but whether the model preserves a consistent thermodynamic balance under the operations that arise naturally in transport analysis: restriction to subdomains, coupling across interfaces, linearization near equilibrium, and discretization for computation.
This paper develops a continuum-first framework for open nonlinear convection--diffusion systems in which thermodynamic consistency is formulated as a free-energy balance with nonnegative bulk dissipation and explicit boundary and source contributions. Within this setting, nonlinear transport systems are defined as structured objects built from admissible state fields, storage functionals, constitutive flux decompositions, sources, and boundary ports. We prove that the thermodynamic balance is preserved under exact structure-preserving transformations, restriction to subdomains, local-to-global reconstruction over compatible domain decompositions, and power-conserving interconnection of open subsystems. We then derive classical linear convection--diffusion models as tangent thermodynamic descendants at equilibrium and show that the same invariant survives weak formulation, semidiscretization, and fully discrete time stepping when the numerical design respects thermodynamic structure. Nonlinear drift--diffusion and porous-medium convection--diffusion are used as explicit examples. The resulting contribution is a compositional transport framework in which the second law remains visible across continuum modeling, subsystem coupling, linear approximation, and computation.