A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems
- Mielke, Alexander
2010 Mathematics Subject Classification
- 35K45 35Q92 35Q80
- Gradient structure, reversible mass action reactions, free energy functional, entropy functional, energy balance equation, van Roosbroeck equations
In recent years the theory of Wasserstein distances has opened up a new treatment of the diffusion equations as gradient systems, where the entropy takes the role of the driving functional and where the space is equipped with the Wasserstein metric. We show that this structure can be generalized to closed reaction-diffusion systems, where the free energy (or the entropy) is the driving functional and further conserved quantities may exists, like the total number of chemical species. The metric is constructed by using the dual dissipation potential, which is a convex function of the chemical potentials. In particular, it is possible to treat diffusion and reaction terms simultaneously. The same ideas extend to semiconductor equations involving the electron and hole densities, the electrostatic potential, and the temperature.
- Nonlinearity, 24 (2011) pp. 1329--1346.