Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity

Abstract

We consider the chemotaxis-fluid system          n t + u · ∇n = ∇ · (D(n)∇n) − ∇ · (nS(x, n, c) · ∇c), c t + u · ∇c = ∆c − nf (c),in a bounded convex domain Ω ⊂ R 3 with smooth boundary, where φ ∈ W 1,∞ (Ω) and D, f and S are given functions with values in [0, ∞), [0, ∞) and R 3×3 , respectively.In the existing literature, the derivation of results on global existence and qualitative behavior essentially relies on the use of energy-type functionals which seem to be available only in special situations, necessarily requiring the matrix-valued S to actually reduce to a scalar function of c which, along with f , in addition should satisfy certain quite restrictive structural conditions.The present work presents a novel a priori estimation method which allows for removing any such additional hypothesis: Besides appropriate smoothness assumptions, in this paper it is only required that f is locally bounded in [0,for all n ≥ 0 with some k D > 0 and someIt is shown that then for all reasonably regular initial data, a corresponding initial-boundary value problem for (0.1) possesses a globally defined weak solution.The method introduced here is efficient enough to moreover provide global boundedness of all solutions thereby obtained in that, inter alia, n ∈ L ∞ (Ω × (0, ∞)). Building on this boundedness property, it can finally even be proved that in the large time limit, any such solution approaches the spatially homogeneous equilibrium (n 0 , 0, 0) in an appropriate sense, where n 0 := 1 |Ω| Ω n 0 , provided that merely n 0 ≡ 0 and f > 0 on (0, ∞). To the best of our knowledge, these are the first results on boundedness and asymptotics of large-data solutions in a three-dimensional chemotaxisfluid system of type (0.1).

Existence of global solutions for a chemotaxis-fluid system with nonlinear diffusion

“…In case that α > 1/6, the bounded weak solutions can be constructed with the assumptions that χ, κ satisfy χ ′ ∈ L ∞ loc and κ ∈ L ∞ loc with κ(·) ≥ 0 and κ(0) = 0 (compare to [16] for bounded domain case). If we assume further that either…”

“…For the case of R 3 , [3] proved that global bounded weak solutions of (KSS) exist when α > 1/4 under the hypothesis either χ ′ (·) ≥ χ 0 > 0 or κ ′ (·) ≥ κ 0 > 0, where χ 0 and κ 0 are positive constants. It was shown, very recently, in [16] that in case that α > 1/6, bounded weak solutions are constructed in a bounded convex domain Ω ⊂ R 3 with smooth boundary with more relaxed conditions on χ and κ in general setting (see [16,Theorem 1.1] for the details).…”

Global existence and boundedness in a higher-dimensional quasilinear chemotaxis system

“…Our method is motivated by Winkler [36], where the author proved that for m > 7 6 and n = 3, the solution of system (1.1) posed on a bounded convex domain is global and bounded as a special case of his results. Our result is consistent with that of [36] when n = 3.…”

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

“…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]).…”