Section: New Results

Existence of very weak global solutions to cross diffusion models

The entropy structure has been used in [26] to derived a very general theorem for existence for cross diffusion models. The theory is based on the interplay between the entropy structure which gives some compactness in space (gradient control) and the duality strucutre identified by Michel Pierre for general parabolic systems, which gives integrability. We derive a very general results under very general structural hypothesis (exsitence of an entropy which is compatible with reaction terms and relevance of the duality strucutre). The key is the construction of implicit solutions of the semi discete version (time is discretized) which happens to verify all the strucutres and are very regular. Moreover, we give a simple condition for mutliple case (more than 3 species) for building examples with an entropy structure based on the detailed balance structure proposed in [34].