Dynamics in an attraction-repulsion Navier-Stokes system with signal-dependent motility and sensitivity

This paper is concerned with an attraction-repulsion Navier-Stokes system with signal-dependent motility and sensitivity in a two-dimensional smooth bounded domain under zero Neumann boundary conditions for n, c, v and the homogeneous Dirichlet boundary condition for u. This system describes the evolution of cells that react on two different chemical signals in a liquid surrounding environment and models a density-suppressed motility in the process of stripe pattern formation through the self-trapping mechanism. The major difficulty in analysis comes from the possible degeneracy of diffusion as c and v tend to infinite. Based on a new weighted energy method, it is proved that under appropriate assumptions on parameter functions, this system possesses a unique global classical solution, which is uniformly-in-time bounded. Moreover, by means of energy functionals, it is shown that the global bounded solution of the system exponentially converges to the constant steady state.

Automated summary

This paper discusses an attraction-repulsion Navier-Stokes system with signal-dependent motility and sensitivity in a two-dimensional smooth bounded domain, modeling cells' reaction to two chemical signals in a liquid environment and density-suppressed motility during stripe pattern formation. Through a new weighted energy method, it is proven that under certain assumptions, the system has a unique global classical solution that is uniformly bounded in time, and the global solution exponentially converges to a constant steady state.