Sharp interface limit in a phase field model of cell motility

par Leonid Berlyand (Pennsylvania State University)

mardi 8 déc. 2015, 13:00 → 14:00 Europe/Paris

salle 112 (ICJ, UCBL - La Doua, Bât. Braconnier)

We consider a system of two parabolic PDEs arising in modeling of motility of eukaryotic cells on substrates. The two key properties of this system are (i) presence of gradients in the coupling terms (gradient coupling) and (ii) mass (volume) preservation constraints. We derive the equation of the motion of the cell boundary, which is the mean curvature motion perturbed by a novel nonlinear term and prove that the sharp interface property of initial conditions is preserved in time. We next show that this novel term leads to surprising features of the motion of the interface such as discontinuities of the interface velocity and hysteresis. Because of the properties (i)-(ii), classical comparison principle techniques do not apply to this system. Furthermore, the system can not be written in a form of gradient flow, which is why recently developed Gamma-convergence techniques also can not be used. A special form of asymptotic expansion is introduced to reduce analysis to a single nonlinear PDE: a one-dimensional model problem. Stability analysis reveals a qualitative change in the behavior of the system depending on the main physical parameter. This is joint work with V. Rybalko and M. Potomkin.