Global classical small-data solutions for a three-dimensional chemotaxis Navier–Stokes system involving matrix-valued sensitivities

Abstract

The coupled chemotaxis fluid system

Long-term behaviour in a chemotaxis-fluid system with logistic source

“…With this as starting point, we could follow the reasoning of [41, Lemma 3.9] to derive a contradiction to (5). There differential inequalities for Ω n 4 ε and Ω |A 1 2 u ε | 2 first yielded bounds for these quantities on [0, T max,ε ), then smoothing estimates for the Stokes semigroup (if combined with an embedding for the domains of fractional powers of A) and for the Neumann heat semigroup led to estimates for u ε L ∞ (Ω×(0,Tmax,ε)) , ∇c ε L ∞ ((0,Tmax,ε),L 4 (Ω)) and n ε L ∞ (Ω×(0,Tmax,ε)) .…”

“…Other variants of the model that are commonly treated include nonlinear (porous medium type) diffusion of bacteria, where ∆n is replaced by ∆n m for some m > 1 (see [34,35,10,7,49]), thereby improving chances for finding bounded solutions, or, exchanging χ∇ · (n∇c) for ∇ · (nS(n, c, x)∇c), more complex sensitivity functions S ( [46,40,39,16,5]), which may be matrix-valued, thus introducing new mathematical challenges by destroying the natural energy structure of the system and, seen from the biological viewpoint, taking care of more complicated swimming behaviour of bacteria (cf. [9,28,47]).…”

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

“…Proof The outcome of this lemma can be achieved by utilizing regularity estimates for the Stokes semigroup and embedding properties for domains of fractional powers of the Stokes operator (cf, eg, Cao and Lankeit, Lemma 2.3). Detailed proofs can be found in Wang and Xiang, Lemma 2.4 and 2.5 ( N = 2), and Winkler , Corollary ( N = 3).…”

Global weak solutions and eventual smoothness in a 3D two‐competing‐species chemotaxis‐Navier‐Stokes system with two consumed signals

“…Furthermore, Winkler proved that under the explicit hypothesis of damping coefficient, these solutions are shown to stabilize toward a spatially homogeneous state. For more related works about Keller‐Segel chemotaxis‐fluid systems, please see other studies . On the other hand, the fully parabolic two‐species chemotaxis‐competition system without fluid has been studied by some authors (see Wang et al, Lin et al, Bai and Winkler).…”