Global existence and boundedness in a Keller–Segel–Stokes system involving a tensor-valued sensitivity with saturation

Abstract

This paper deals with a boundary-value problem in two-dimensional smoothly bounded domains for the coupled Keller-Segel-Stokes systemHere, one of the novelties is that the chemotactic sensitivity S is not a scalar function but rather attains values in R 2×2 , and satisfies |S(x, n, c)| ≤ C S (1 + n) −α with some C S > 0 and α > 0. We shall establish the existence of global bounded classical solutions for arbitrarily large initial data. In contrast to the corresponding case of scalar-valued sensitivities, this system does not possess any gradient-like structure due to the appearance of such matrix-valued S. To overcome this difficulty, we will derive a series of a priori estimates involving a new interpolation inequality.To the best of our knowledge, this is the first result on global existence and boundedness in a KellerSegel-Stokes system with tensor-valued sensitivity, in which production of the chemical signal is involved.

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

“…= Ω n ε (t) is bounded according to Lemma 2.5, this entails (40). The estimate in (41) also shows that…”

“…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]).…”

“…[9,28,47]). In contrast to (1), in the classical Keller-Segel system the chemoattractant is produced by the bacteria themselves and not consumed (accounting for terms +n−c in place of −nc in the second equation of (1)), and models of Keller-Segel-Stokes type have also been considered ( [40,3]). In Ω = R 3 , mild solutions to a system encompassing both mechanisms at the same time were proven to exist under a smallness condition on initial data ( [20]).…”

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). □…”

Singular sensitivity in a Keller–Segel-fluid system

“…Arguments appearing in the proof of the lemma below have been previously used in e.g. [41] and [35] and rely on semigroup estimates for the Stokes semigroup. …”