Traveling wave for a coupled incompressible Darcy's free boundary model with undercooling effect and surface tension

We present a two-dimensional Hele-Shaw free-boundary model for the motility of eukaryotic cells on substrates. The key ingredients of this model are Darcy's law for the over-damping motion of the cytoplasm as a confined viscous droplet with particular boundary consition for the Young-Laplace equation to describe a cytoskeleton-induced active force that can be generated in cells either by actin polymerization against the membrane, or by actomyosin contraction of cortical filaments, which adhere to the membrane, and friction of the membrane on the substrate. First, we study the linear stability of the steady state and prove that above a threshold, the disk is linearly unstable. This analysis highlights the stabilizing effect of undercooling. Then, using a bifurcation argument, we prove the existence of traveling waves that describe a persistent motion in cell migration and justify the relevance of the model.