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

TL;DR

This paper proves the existence of traveling wave solutions in a complex free boundary model of cell motility, demonstrating persistent cell motion through nonlinear analysis and bifurcation techniques.

Contribution

It introduces new mathematical methods to establish traveling wave solutions in a nonlinear free boundary problem modeling cell motility.

Findings

Disk becomes linearly unstable above a certain threshold.

Existence of explicit traveling wave solutions is demonstrated.

Traveling waves describe persistent biological cell motion.

Abstract

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.