Global Large-Data Solutions in a Chemotaxis-(Navier–)Stokes System Modeling Cellular Swimming in Fluid Drops

Abstract

In the modeling of collective effects arising in bacterial suspensions in fluid drops, coupled chemotaxis-(Navier–)Stokes systems generalizing the prototype have been proposed to describe the spontaneous emergence of patterns in populations of oxygen-driven swimming bacteria. Here, κ ∈ ℝ and the gravitational potential φ are given and Ω ⊂ ℝ N is a bounded convex domain with smooth boundary. Under the boundary conditions and u = 0 on ∂Ω, it is shown in this paper that suitable regularity assumptions on the initial data entail the following: • If N = 2, then the full chemotaxis-Navier–Stokes system (with any κ ∈ ℝ) admits\\p0 a unique global classical solution. • If N = 3, then the simplified chemotaxis-Stokes system (with κ = 0) possesses at\\p0 least one global weak solution. In particular, no smallness condition on either φ or on the initial data needs to be fulfilled here, as required in a related recent work by Duan et al. [5].