Weierstraß-Institut für Angewandte Analysis und Stochastik ÖÖÖÖÖÒØ ×ØÖÙ ØÙÖÖ ÓÖ ÖÖÖ ØØÓÒ¹¹¹¹Ù××ÓÒ ×Ý×ØØÑ× ÒÒ ÓÖ ÒÒÖÖݹ¹ÖÖÖع¹¹¹Ù××ÓÒ ×Ý×ØØÑ× ØØ ØÓ ÀÀÖÖÖØ Û××× ÓÒ ØØØ Ó ××ÓÒ ÓÓ × ¼ØØ ÖØØØØÝ

Im Forschungsverbund, Berlin Preprint
unpublished
×ÙÙÑÑØØØØØ ÅÅÖ¸¾¼½¼ÅÅÖ¸ÅÅÖ¸¾¼½¼ 1 ÏÖ×ØÖÖ×× ÁÒ×ØØØÙØØ ÓÖ ÔÔÐÐÐÐ ÒÒÐÝ××× ÒÒ ËØÓ×ØØ× ÅÓÓÖÖÒ×ØÖº ¿¿ ½¼½½½ ÖÐÐÒ ÖÑÑÒÝ ¹ÅÅÅÐÐ ÑÑÑÐÐÛÛÛ×¹¹ÖÐÐÒººº 2 ÁÒ×ØØØÙØ Ö ÅÅØØØÑÑØØØ ÀÙÑÓÐÐعÍÒÒÚÖ××ØØØ ÞÙ ÖÐÐÒ ÊÙÙÓÛÖ Ù×××× ¾¾ ½¾¾¾¾ ÖÐÐÒ¹¹ÐÐÖ××ÓÓ ÖÑÑÒÝ AEÓº ½½½½ ÖÐÐÒ ¾¼½¼ 2010 Mathematics Subject Classification. 35K45, 35Q92, 35Q80. Key words and phrases. Gradient structure, reversible mass action reactions, free energy functional, entropy functional, energy balance equation, van Roosbroeck equations. Research
more » ... ions. Research partially supported by DFG via SFB 787 Semiconductor Nanophotonics: Materials, Models, Devices, project B4 "Multi-dimensional modeling and simulation of VCSELs". ØØØ Ý ÏÖ×ØÖÖÖ¹ÁÒ×ØØØÙØ Ö ÒÒÒÛÒÒØØ ÒÒÐÝ××× ÙÒÒ ËØÓ×ØØØ´ÏÁÁ˵×ØØØ´ÏÁÁ˵ ÅÓÓÖÖÒ×ØÖÖÖÖ ¿¿ ½¼½½½ ÖÐÐÒ ÖÑÑÒÝ ÜÜ · ¿¼ ¾¼¼¼¼¼¼ ¹ÅÅÅÐÐ ÔÖÖÔÖÖÒØÛÛÛ×¹¹ÖÐÐÒººº ÏÓÖÐÐ ÏÏÏÏ Ï ØØÔÔ»»ÛÛÛºÛÛÛ×¹¹ÖÐÐÒººº» Abstract In recent years the theory of the Wasserstein metric has opened up a new treatments of the diffusion equations as gradient systems, where the free energy or the entropy take the role of the driving functional and where the space is equipped with the Wasserstein metric. We show on the formal level that this gradient structure can be generalized to reaction-diffusion systems with reversible mass-action kinetic. The metric is constructed by using the dual dissipation potential, which is a quadratic functional of all chemical potentials including the mobilities as well as the reaction kinetics. The metric structure is obtained by Legendre transform from the dual dissipation potential. The same ideas extend to systems including electrostatic interactions or a correct energy balance via coupling to the heat equation. We show this by treating the semiconductor equations involving the electron and hole densities, the electrostatic potential, and the temperature. Thus, the models in [AGH02], which stimulated this work, have a gradient structure.
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