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

• Published in: Communications in partial differential equations Vol. 37; no. 2; pp. 319 - 351
• Main Author: Winkler, Michael
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis Group 01.02.2012; Taylor & Francis Ltd
• Subjects: A priori estimates; Chemotaxis; Global existence; Navier-Stokes; Partial differential equations; Stokes; Studies
• ISSN: 0360-5302; 1532-4133
• DOI: 10.1080/03605302.2011.591865

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 ].