A Keller‐Segel‐fluid system with singular sensitivity: Generalized solutions

Abstract

In bounded smooth domains normalΩ⊂RN, N ∈ {2,3}, we consider the Keller‐Segel‐Stokes system nt+u·∇n=normalΔn−χ∇·()nc∇c,ct+u·∇c=normalΔc−c+n,ut=normalΔu+∇P+n∇ϕ,2em∇·u=0, and prove global existence of generalized solutions if χ<∞,N=2,53,N=3. These solutions are such that blow‐up into a persistent Dirac‐type singularity is excluded.

Stationary solutions to a chemotaxis-consumption model with realistic boundary conditions for the oxygen

“…Thus, c solves (8) fulfilling m = Ω n = α Ω e c = m(α). Uniqueness of α satisfying m(α) = m and of the solution of (8) with this value of α (according to Lemma 3.8) show uniqueness of the solution to (4). That this solution is not constant for Ω n > 0 implying α > 0 can be seen from Lemma 3.1.…”

“…One of the main questions motivating the study of this system and its relatives was whether and how (2) can account for the emergence of large scale coherent patterns, as observed experimentally in [11,38]. There were also other motivations; see, e.g., the question posed in the title of [44], or the introduction of [4] and references therein; but at least with regard to the first-mentioned matter, results on the long-term behaviour of solutions to (2), paint a different picture: Not only small-data solutions to (2) in three-dimensional domains (see e.g. [6] or [7,45,35]), but also every classical solution in Ω ⊂ R 2 ([41, 46, 17, 12]) and even every "eventual energy solution" to (2) converges to the stationary, constant state ( 1 |Ω| Ω n 0 , 0, 0), [44].…”

“…We now want to employ the information on the scalar equation (8) obtained in the previous section for solving the actual system (4). In order to make sure that each can be transformed into the other, we look at the first equation of (4):…”

“…Proof of Theorem 1.1. Due to Lemma 3.16, there is exactly one number α ∈ [0, ∞) such that m(α) = m. With this α, Lemma 3.6 ensures solvability of (8), and setting n := αe c , we obtain a solution to (4) with Ω n = m(α).…”