Convergence rates of solutions for a two-dimensional chemotaxis-Navier-Stokes system

By Qingshan Zhang, Yuxiang Li. 2015

116 citations recorded by the source corpus.

We consider an initial-boundary value problem for the incompressible chemotaxis-Navier-Stokes equation \begin{eqnarray*} \left\{\begin{array}{lll} n_t + u \cdot \nabla n = \Delta n - \chi\nabla\cdot(n \nabla c),&{} x\in\Omega,\ t>0,\\ c_t + u \cdot \nabla c = \Delta c - nc, &{} x \in \Omega,\ t>0,\\ u_t + \kappa(u\cdot\nabla)u = \Delta u + \nabla P + n\nabla\phi ,&{} x\in\Omega,\ t>0,\\ \nabla\cdot u=0, &{}x\in\Omega,\ t>0, \end{array}\right. \end{eqnarray*} in a bounded domain $\Omega\subset\mathbb{R}^2$. It is known that if $\chi>0$, $\kappa\in\mathbb{R}$ and $\phi\in C^2(\bar{\Omega})$, for sufficiently smooth initial data, the model possesses a unique global classical solution which satisfies $(n, c, u)\rightarrow(\bar{n}_0, 0, 0)$ as $t\rightarrow\infty$ uniformly with respect to $x\in\Omega$, where $\bar{n}_0:=\frac{1}{|\Omega|}\int_{\Omega}n(x, 0)dx$. In the present paper, we prove this solution converges to $(\bar{n}_0, 0, 0)$ exponentially in time.

Subfields: Cell Biology, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/3978a0b381a401b78eb4ce547b3b30286008b7fa

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Citations in the corpus

A three-dimensional Keller–Segel–Navier–Stokes system with logistic source: Global weak solutions and asymptotic stabilization (2019, background): “For such models, numerous results have been achieved in the past few years, addressing topics ranging from the mere existence of global solutions ([13], [73], [3], [6]), also in the context of nonlinear cell diffusion ([14], [55], [60], [9], [77]), to global boundedness ([66], [5], [24], [4]) and ev”
Long-term behaviour in a chemotaxis-fluid system with logistic source (2016, background): “In the setting of [44], the convergence of solutions to the stationary state was proven in [45]; its rate was given in [48].”
Depleting the signal: Analysis of chemotaxis‐consumption models—A survey (2023, background): “6 ([162]) Let n ≥ 1 and Ω ⊂ R be a bounded domain with smooth boundary, and suppose that for some T > 0, the pair (u, v) ∈ ( C2,1(Ω × [T,∞)) )2 solves the boundary value problem in (CC1) classically and is such that (2.”
Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity (2018, background): “2) ([54], [64]), and also extends known facts on regularity and large time behavior in the associated fluid-free analogue 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”