The stability analysis of a two-dimensional Keller-Segel-Stokes system

In a bounded domain of Ω⊂R ²with smooth boundary, the homogeneous Neumann-Neumann-Dirichlet initial-boundary-value problem of a class of uncompressible Keller-Segel-Stokes equations system has been studied, and the global existence and boundedness of the classical solutions to this problem have been constructed under the appropriate smallness condition on the initial value of n ₀. Thus what's next is to study the long-time behavior of the classical solutions based on the condition above, it is proven that the classical solutions to the problem converge to the spatial equilibrium at exponential rate as t goes to infinity under the extra smallness assumption on the initial data n ₀, where then n ₀stands for the integral mean value of nover Ω, and the gravitational potential φbelongs to W ^2,∞(Ω).