Global weak solvability in a self-consistent chemotaxis-Navier–Stokes system involving Dirichlet boundary conditions for the signal.

The self-consistent chemotaxis-Navier–Stokes system with nonlinear diffusion ∂ t n + u ⋅ ∇ n = ∇ ⋅ (n m − 1 ∇ n) − ∇ ⋅ (n ∇ c) + ∇ ⋅ (n ∇ ϕ) , ∂ t c + u ⋅ ∇ c = Δ c − n c , ∂ t u + (u ⋅ ∇) u + ∇ P = Δ u − n ∇ ϕ + n ∇ c , ∇ ⋅ u = 0 is considered in a bounded domain Ω ⊂ ℝ 2 with smooth boundary. Compared to the previously most-studied chemotaxis-fluid system proposed in [I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA.102 (2005) 2277–2282], the coupling in this system is stronger and more nonlinear. When the system is accompanied by homogeneous boundary conditions of no-flux type for n and c , and of Dirichlet type for u , a quasi-Lyapunov structure provides sufficient regularity features to facilitate a basic existence theory. However, if we change the boundary condition of the signal to c = c ⋆ (x , t) , x ∈ ∂ Ω ,   t > 0 , with a given non-negative function c ⋆ ∈ C 2 (Ω ̄ × [ 0 , ∞)) , then the Dirichlet boundary condition imposed here seems to destroy the quasi-Lyapunov structure. Despite this, we shall find a new energy structure and prove that for suitably regular initial data, the assumption m > 1 is sufficient for the global existence and boundedness of the weak solution. To the best of our knowledge, this is the first work on the global well-posedness problem of the self-consistent chemotaxis-fluid system involving Dirichlet boundary conditions for the signal. [ABSTRACT FROM AUTHOR]