Existence of Smooth Solutions to Coupled Chemotaxis-Fluid Equations

By Myeongju Chae, Kyungkeun Kang, Jihoon Lee. 2011

217 citations recorded by the source corpus.

We consider a system coupling the parabolic-parabolic Keller-Segel equations to the in- compressible Navier-Stokes equations in spatial dimensions two and three. We establish the local existence of regular solutions and present some blow-up criteria. For two dimensional Navier-Stokes-Keller-Segel equations, regular solutions constructed locally in time are, in reality, extended globally under some assumptions pertinent to experimental observation in [20] on the consumption rate and chemotactic sensitivity. We also show the existence of global weak solutions in spatially three dimensions with rather restrictive consumption rate and chemotactic sensitivity.

Subfields: Biomedical Engineering, Cell Biology, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/7fc4773644fc8def63fa93cbc486b0b2d0b09056

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Biomedical Engineering

In Cell Biology

In Modeling and Simulation

Citations in the corpus

A three-dimensional Keller–Segel–Navier–Stokes system with logistic source: Global weak solutions and asymptotic stabilization (2019, background): “( n(·, t), c(·, t) ) → (ρ+ μ , ρ+ μ ) in L(1)(Ω)× L(Ω) for all p ∈ [1, 6) as t→ ∞.”
Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity (2018, background): “also [5] and [65] for recent results on the whole space R2), whereas in the corresponding three-dimensional analogue at least certain global”
Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity (2015, background): “Alternative structural conditions can be found in the more recent work [3], in which for the Navier-Stokes variant of (1.”
Global classical small-data solutions for a three-dimensional chemotaxis Navier–Stokes system involving matrix-valued sensitivities (2016, background): “Due to the difficulties associated with the Navier-Stokes equations in three-dimensional domains, many of these works focus on the two-dimensional case ([60, 59, 46, 68, 66]) or more favorable variants of the model, for example by resorting to the Stokes equation upon neglection of the nonlinear con”
Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with nonlinear diffusion (2015, background): “1) in the whole space we refer the reader to [2, 3, 6, 13,31,34].”
Global existence and boundedness in a 3D Keller–Segel–Stokes system with nonlinear diffusion and rotational flux (2017, background): “Despite the known challenges arising from the theory of fluid equations and the study of chemotaxis systems, in the past several years there have been numerous analytical approaches that addressed issues of well-posedness for corresponding initial (boundary) value problems in either bounded or unbou”
Blow-up prevention by quadratic degradation in a two-dimensional Keller–Segel–Navier–Stokes system (2016, background): “Then still the understanding of the associated chemotaxis-only subsystem is far from complete especially in the case N = 3 ([29]), but the dampening effect of signal absorption at least under certain restrictions on the space dimension and the model parameter functions allows for the construction of”
Stabilization in a two-dimensional chemotaxis-Navier–Stokes system (2013, unknown): “1) with large initial data can be found in [3] and [32].”