Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion

By Youshan Tao, Michael Winkler. 2012

215 citations recorded by the source corpus.

This paper deals with a boundary-value problem in two-dimensional smoothly bounded domains for the coupled chemotaxis-fluid model $$ \left\{ \begin{array}{l} n_t+ u\cdot \nabla n=\Delta n^m - \nabla \cdot (n\chi(c)\nabla c)\\ c_t+ u\cdot \nabla c=\Delta c-nf(c)\\ u_t +\nabla P-\eta \Delta u+n \nabla \phi=0 \\ \nabla \cdot u=0, \end{array} \right. $$ which describes the motion of oxygen-driven swimming bacteria in an incompressible fluid. The given functions $\chi$ and $f$ are supposed to be sufficiently smooth and such that $f(0)=0$. It is proved that global bounded weak solutions exist whenever $m>1$ and the initial data $(n_0,c_0,u_0)$ are sufficiently regular satisfying $n_0 \ge 0$ and $c_0\ge 0$. This extends a recent result by Di Francesco, Lorz and Markowich (Discrete Cont. Dyn. Syst. A 28 (2010)) which asserts global existence of weak solutions under the constraint $m \in (\frac{3}{2},2]$.

Subfields: Computational Theory and Mathematics, Finance, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/bf6ba6505b1a5485d9027133fcf4994f73f3c339

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Citations in the corpus

How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system? (2015, background): “In the case m > 43 , global existence of – possibly unbounded – weak solutions has been established in [33] even for the associated full chemotaxis-Navier-Stokes system; results on global existence and boundedness in two-dimensional chemotaxis-fluid systems with nonlinear cell diffusion can be found”
Existence of Smooth Solutions to Coupled Chemotaxis-Fluid Equations (2011, background): “In [20], Tao and Winkler extended the result to the case m > 1 on a bounded domain in R(2).”
Stabilization in a two-dimensional chemotaxis-Navier–Stokes system (2013, unknown): “or of additional logistic terms, can be found in [5], [30], [34], [1] and [14], for instance, and recently statements on global existence and boundedness have been derived in [6] and [28] for a two-dimensional chemotaxis-Stokes variant of (1.”
Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity (2018, background): “2) obtained on replacing ∆n by nonlinear diffusion operators generalizing the porous medium-type choice ∆n for several ranges of m > 1 ([10], [33], [12], [48], [26], [43], [44], [56]).”
Blow-up prevention by quadratic degradation in a two-dimensional Keller–Segel–Navier–Stokes system (2016, background): “nonlinear diffusion ([5], [33], [30], [31], [8]) and variants in the crossdiffusive term ([1], [34], [16]).”
Long-term behaviour in a chemotaxis-fluid system with logistic source (2016, background): “Other variants of the model that are commonly treated include nonlinear (porous medium type) diffusion of bacteria, where ∆n is replaced by ∆n 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 co”
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”
Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion (2013, background): “This global existence result in the spatially two-dimensional setting could recently be extended in [22] so as to cover the whole range m ∈ (1,∞), and moreover it has been shown there that all solutions evolving from sufficiently regular initial data are uniformly bounded in Ω ×(0,∞).”
The small-convection limit in a two-dimensional chemotaxis-Navier–Stokes system (2017, unknown): “1) ([7], [6], [21], [33], [34], [35], [29]).”