Singular sensitivity in a Keller–Segel-fluid system

Abstract

Abstract. In bounded smooth domains Ω ⊂ R N , N ∈ {2, 3} considering the chemotaxis-fluid systemwith singular sensitivity, we prove global existence of classical solutions for given φ ∈ C 2 (Ω), for κ = 0 (Stokes-fluid) if N = 3 and κ ∈ {0, 1} (Stokes-or NavierStokes fluid) if N = 2 and under the condition thatMSC (

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

“…In the presence of fluid coupling, we have been able to obtain global existence of classical solutions for < √ 2 N in Black et al 2 (We also refer to the introduction of said article for additional motivation and more references to works dealing with chemotaxis-fluid systems or chemotaxis systems with logarithmic sensitivity.) This parameter range for is (almost) as large as that known for the fluid-free system (cf previous studies [3][4][5][6] )-there it is only known to be slightly larger in N = 2, cf Lankeit.…”

“…In order to see how, in the latter setting, reliance on estimates for presents us with a problem, let us recall the main difficulty stemming from presence of the fluid coupling in Black et al:…”

“…In Black et al,, lemma 2.5 we mitigated this problem by replacing the use of semigroup estimates by an argument based on the differential inequality (Black et al, lemma 2.4) for arbitrary q > 1, where we were able to control the source term mainly due to the bound on previously obtained ,. lemma 2.3 This will no longer be possible if p < 1.…”

“…Unfortunately, this reasoning fails for N = 3 (cf Remark ). Here, we instead employ a differential inequality for – for q < 1, in contrast to Black et al One of its consequences is a bound on the space‐time integral of (Lemma ), which we then use to secure bounds for in Lemma for . (This reasoning, in turn, would work for N = 2 but entail some restrictions on χ .…”

“…In the two-dimensional setting, we firstly procure bounds for the fluid velocity field and then rely on the well-known smoothing estimates for the heat semigroup; however, we need more than a straightforward application and have to partially absorb the additional source term by the term to be controlled.Unfortunately, this reasoning fails for N = 3 (cf Remark 3.8). Here, we instead employ a differential inequality for d dt ∫ Ω c q -for q < 1, in contrast to Black et al 2 One of its consequences is a bound on the space-time integral of | | | ∇c q 2 | | | 2 (Lemma 3.9), which we then use to secure bounds for ∫ T 0 ∫ Ω c r in Lemma 3.10 for r ∈…”