  We consider motility of keratocyte cells driven by myosin contraction and
introduce a 2D free boundary model for such motion. This model generalizes a 1D
model from [12] by combining a 2D Keller-Segel model and a Hele-Shaw type
boundary condition with the Young-Laplace law resulting in a boundary curvature
term which provides a regularizing effect. We show that this model has a family
of traveling solutions with constant shape and velocity which bifurcates from a
family of radially symmetric stationary states. Our goal is to establish
observable steady motion of the cell with constant velocity. Mathematically,
this amounts to establishing stability of the traveling solutions. Our key
result is an explicit asymptotic formula for the stability-determining
eigenvalue of the linearized problem. This formula greatly simplifies the task
of numerically computing the sign of this eigenvalue and reveals the physical
mechanisms of stability. The derivation of this formula is based on a special
ansatz for the corresponding eigenvector which exhibits an interesting singular
behavior such that it asymptotically (in the small-velocity limit) becomes
parallel to another eigenvector. This reflects the non-self-adjoint nature of
the linearized problem, a signature of living systems. Finally, our results
describe the onset of motion via a transition from unstable radial stationary
solutions to stable asymmetric traveling solutions.
