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We consider a model of congestion dynamics with chemotaxis, where the density of cells follows the chemical signal it generates, while observing an incompressibility constraint (incompressible parabolic-elliptic Patlak-Keller-Segel model). We show that when the chemical diffuses slowly and attracts the cells strongly, then the dynamics of the congested cells is well approximated by a surface-tension driven free boundary problem. More precisely, we rigorously establish the convergence of the solution to the characteristic function of a set whose evolution is determined by the classical Hele-Shaw free boundary problem with surface tension. The problem is set in a bounded domain, which leads to an interesting analysis on the limiting boundary conditions. Namely, we prove that the assumption of Robin boundary conditions for the chemical potential leads to a contact angle condition for the free interface (in particular Neumann boundary conditions lead to an orthogonal contact angle condition, while Dirichlet boundary conditions lead to a tangential contact angle condition). 

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

We investigate the fractional diffusion approximation of a kinetic equation set in a bounded interval with diffusive reflection conditions at the boundary. In an appropriate singular limit corresponding to small Knudsen number and long time asymptotic, we show that the asymptotic density function is the unique solution of a fractional diffusion equation with Neumann boundary condition. This analysis completes a previous work by the same authors in which a limiting fractional diffusion equation was identified on the half-space, but the uniqueness of the solution (which is necessary to prove the convergence of the whole sequence) could not be established.