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Global existence and stabilization of classical solutions to a Keller-Segel-(Navier-)Stokes system with tensor-valued sensitivity and prescribed signal concentration on the boundary
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DOI：10.1142/S021820252550006X.png)
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Abstract
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This paper investigates a logistic-type Keller-Segel-fluid system with bounded tensor-valued sensitivity in a bounded domain with smooth boundary, subject to no-flux, Dirichlet, and Dirichlet boundary conditions for the cells, chemicals, and fluids, respectively. Unlike previous studies, which primarily focused on scalar chemotactic sensitivity under homogeneous Neumann boundary conditions for the signal concentration, the matrix-type chemotactic sensitivity in this work prevents the offsetting of contributions from cross-diffusive reciprocity. Moreover, the inhomogeneous Dirichlet boundary conditions introduce additional integral terms on both the boundary and the prescribed boundary function during the process of a priori estimation, presenting significant mathematical challenges. By fully leveraging the quadratic damping effects and key system features, it is shown that in the two-dimensional case, the system admits a globally uniformly bounded classical solution for any suitably regular initial data, even when the Navier-Stokes fluid is involved. In the three-dimensional case, a similar conclusion holds for the system coupled with the Stokes fluid, provided that the logistic damping is suitably strong. Furthermore, the existence of spatially homogeneous equilibria, as the large-time limit of each solution, has been confirmed under certain additional assumptions.
Keywords:
Keller-Segel-(Navier-)Stokes system
Dirichlet boundary conditions for signal
quadratic degradation
global boundedness
stabilization
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