Some remarks about the well-posedness of lifshitz-slyozov's equations with nucleation kinetics

Juan Calvo (1) , Erwan Hingant (2) , Romain Yvinec (3, 4)

1 UGR - Universidad de Granada = University of Granada
2 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352
3 PRC - Physiologie de la reproduction et des comportements [Nouzilly]
4 MUSCA - Dynamiques de populations multi-échelles pour des systèmes physiologiques

The Lifshitz-Slyozov model is a nonlocal transport equation that can describe certain types of phase transitions in terms of the temporal evolution of a mixture of monomers and aggregates. Most applications of this model so far do not require boundary conditions. However, there is a recent interest in situations where a boundary condition might be needed-e.g. in the context of protein polymerization phenomena. Actually the boundary condition may change dynamically in time, depending on an activation threshold for the monomer concentration. This new setting poses a number of mathematical difficulties for which the existing literature is scarce. In this contribution we construct examples of solutions for which the boundary condition becomes activated (resp. deactivated) dynamically in time; we also discuss how to approach the well-posedness problem for such situations.

Domaines: Equations aux dérivées partielles [math.AP]

Juan Calvo, Erwan Hingant, Romain Yvinec. Some remarks about the well-posedness of lifshitz-slyozov's equations with nucleation kinetics. 2023. ⟨hal-04099684⟩