Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement

Abstract

A class of chemotaxis-Stokes systems generalizing the prototypeis considered in bounded convex three-dimensional domains, where φ ∈ W 2,∞ (Ω) is given.The paper develops an analytical approach which consists in a combination of energy-based arguments and maximal Sobolev regularity theory, and which allows for the construction of global bounded weak solutions to an associated initial-boundary value problem under the assumption that m > 9 8 .(0.1) Moreover, the obtained solutions are shown to approach the spatially homogeneous steady state ( 1 |Ω| Ω n 0 , 0, 0) in the large time limit. This extends previous results which either relied on different and apparently less significant energytype structures, or on completely alternative approaches, and thereby exclusively achieved comparable results under hypotheses stronger than (0.1).

Global boundedness and decay property of a three-dimensional Keller–Segel–Stokes system modeling coral fertilization

“…It has also been observed that when S = S(x, ρ, c) is a tensor, the corresponding chemotaxis-fluid system loses some energylike structure, which plays a key role in the analysis of the scalar-valued case. Despite these challenges, some comprehensive results on the global-boundedness and large time behavior of solutions are available in the literature (see [3,19,20,25,27,29,33,34,35] for example). It has been shown that when S = S(x, ρ, c) is a tensor fulfilling |S(x, ρ, c)| ≤ C S (1 + ρ) α for some α > 0 and C S > 0, (1.4) the three-dimensional system (1.3) with µ = 0, κ = 0 admits globally bounded weak solutions for α > 1/2 [27], which is slightly stronger than the corresponding subcritical assumption α > 1/3 for the fluid-free system.…”

On a chemotaxis model with degenerate diffusion: Initial shrinking, eventual smoothness and expanding

“…When m = 1 in (1), namely, the diffusion of bacteria cell are assumed to be random, Tao and Winkler [29] proved that this model admits a global weak solution, and a more interesting fact is that, the weak solution will become smooth after some time. Recently, chemotaxis models featuring a density-dependent diffusion term have drawn great attention from many authors [7,1,28,35,38,18,34]. For this system with the porous medium diffusion (i.e.…”

Global solvability and boundedness to a coupled chemotaxis-fluid model with arbitrary porous medium diffusion

“…The first effort to this 3-D problem is due to the work by Di Francesco et al [3], in which, they obtained the existence of global bounded weak solutions for m in some finite interval, namely m ∈ 7+ √ 217 12 , 2 (approximating to (1.8109, 2]); It was Tao and Winkler [21], in 2013, who established the global existence of locally bounded weak solutions with m belonging to the infinite interval ( 8 7 , +∞). Afterwards, Winkler [24] supplemented the uniform boundedness of solutions for the case m > 7 6 ; Recently, Winkler [25] further improved this result to the case m > 9 8 with Ω being a convex domain. However, as mentioned by Winkler [25], the question of identifying an optimal condition on m ≥ 1 ensuring global boundedness in the three-dimensional version of (1.3) remains an open challenge.…”

“…Afterwards, Winkler [24] supplemented the uniform boundedness of solutions for the case m > 7 6 ; Recently, Winkler [25] further improved this result to the case m > 9 8 with Ω being a convex domain. However, as mentioned by Winkler [25], the question of identifying an optimal condition on m ≥ 1 ensuring global boundedness in the three-dimensional version of (1.3) remains an open challenge.…”

“…At last, we also extend the results to the coupled chemotaxis-Stokes system. Important progresses for chemotaxis-Stokes system with m > 7 6 , m > 8 7 and m > 9 8 have been carried out respectively by [24,21,25], but leave a gap for 1 < m ≤ 9 8 . Our result for chemotaxis-Stokes system supplements part of the gap ( 11 4 − √ 3, 9 8 ).…”