Conditional estimates in three-dimensional chemotaxis-Stokes systems and application to aKeller-Segel-fluid model accounting for gradient-dependent flux limitation

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:281

Article

Winkler, Michael¹ 

[1] Univ Paderborn, Inst Math, Warburger Str 100, D-33098 Paderborn, Germany

关键词: Chemotaxis;    Stokes;    Flux limitation;   

DOI  :  10.1016/j.jde.2021.02.004

来源: Elsevier

【 摘 要 】

This manuscript is concerned with the Keller-Segel-Stokes system {n(t) + u . del n = del n - del.(nF(vertical bar del c vertical bar(2))del c), c(t) + u . del c = Delta c - c + n, u(t) = Delta u+ del P + nP Phi, del.u = 0, (star) under no-flux/no-flux/Dirichlet boundary conditions in smoothly bounded three-dimensional domains, with given suitably regular functions F and Phi. Here in accordance with recent developments in the literature on refined modeling of chemotactic migration, the introduction of suitably decaying F is supposed to adequately account for saturation mechanisms that limit cross-diffusive fluxes near regions of large signal gradients. In the context of such nonlinearities which suitably generalize the prototype given by F(xi) = K-F(1 + xi)(-alpha/2), xi >= 0, with K-F > 0, known results addressing a fluid-free parabolic-elliptic simplification of ( star) have identified the value alpha(c) = 1/2 as critical with regard to the occurrence of blow-up in the sense that some exploding solutions can be found when alpha < 1/2, whereas all suitably regular initial data give rise to global bounded solutions when alpha > 1/2. The intention of the present study consists in making sure that the latter feature of blow-up prevention by suitably strong flux limitation persists also in the more complex framework of the fully coupled chemotaxis-fluid system (star). To achieve this, as a secondary objective of possibly independent interest the manuscript separately establishes some conditional bounds for corresponding fluid fields and taxis gradients in a fairly general setting that particularly includes the subsystem of (star) concerned with the evolution of (c, u, P). These estimates relate respective regularity features to certain integrability properties of associated forcing terms, as in the context of (star) essentially represented by the quantity n. The application of this tool to the specific problem under consideration thereafter facilitates the derivation of a result on global existence of bounded classical solutions to (star) for widely arbitrary initial data actually within the entire range alpha > 1/2, and by means of an argument which appears to be significantly condensed when compared to reasonings pursued in previous works concerned with related problems. (C) 2021 Elsevier Inc. All rights reserved.

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