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.

中文翻译：

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具有信号依赖的运动性和灵敏度的吸引力-排斥Navier-Stokes系统的动力学

本文涉及在n，c，v的零Neumann边界条件和u的齐次Dirichlet边界条件下的二维光滑有界域中具有信号依赖运动性和灵敏度的吸引排斥Navier-Stokes系统。该系统描述了在液体周围环境中对两种不同化学信号反应的细胞的进化过程，并通过自陷机理在条纹图案形成过程中模拟了密度抑制的运动性。分析的主要困难是由于扩散可能退化为c和v趋于无限。基于一种新的加权能量方法，证明了在对参数函数进行适当假设的情况下，该系统具有唯一的全局经典解，且该统一解在时间上有界。此外，通过能量泛函表明，系统的整体有界解呈指数收敛至恒定稳态。