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.