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We study an incompressible Darcy's free boundary problem, recently introducedin [22]. Our goal is to prove the existence of non-trivial traveling wavesolutions and thus validate the interest of this model to describe cellmotility. The model equations include a convection diffusion equation for thepolarity marker concentration and the incompressible Darcy's equation. Themathematical novelty of this problem is the nonlinear destabilizing term in theboundary condition that describes the active character of the cellcytoskeleton. We first study the linear stability of this problem and we showthat, above a well precise threshold, the disk becomes linearly unstable. Byusing two different approaches we prove existence of traveling wave solutions,which describes persistent motion of a biological cell. One is explicit, byconstruction. The other is established implicitly, as the one bifurcating fromstationary solution.

Traveling wave solution for a coupled incompressible Darcy's free boundary problem with surface tension