A density-constrained model for chemotaxis

Abstract We consider a model of congestion dynamics with chemotaxis: the density cells follows chemical signal it generates, while subject to an incompressibility constraint. The constraint results in formation patches , describing regions where maximal has been reached. these can be described by either Hele-Shaw or Richards equation type flow (depending on whether we diffusion pure advection). Our focus this paper is construction weak solutions for problem via variational discrete time scheme JKO type. also establish uniqueness solutions. In addition, make more rigorous connection between incompressible chemotaxis and free boundary problems motion terms associated pressure variable. particular, obtain new characterising variable as solution obstacle prove that advection case dynamic preserves patches.