  We study an incompressible Darcy's free boundary problem, recently introduced
in [22]. Our goal is to prove the existence of non-trivial traveling wave
solutions and thus validate the interest of this model to describe cell
motility. The model equations include a convection diffusion equation for the
polarity marker concentration and the incompressible Darcy's equation. The
mathematical novelty of this problem is the nonlinear destabilizing term in the
boundary condition that describes the active character of the cell
cytoskeleton. We first study the linear stability of this problem and we show
that, above a well precise threshold, the disk becomes linearly unstable. By
using two different approaches we prove existence of traveling wave solutions,
which describes persistent motion of a biological cell. One is explicit, by
construction. The other is established implicitly, as the one bifurcating from
stationary solution.
