This paper is concerned with the boundary layer problem on a chemotaxis-Navier-Stokes system modelling boundary layer formation of aerobic bacteria in fluid. Completing the system with physical Robin-type boundary conditions for oxygen, no-flux and Dirichlet boundary conditions for bacteria and fluid velocity, we show that the gradients of its radial solutions in a region between two concentric spheres possessing boundary layer effects as the oxygen diffusion rate $\varepsilon$ goes to zero and the boundary-layer thickness is of order $\mathcal{O}(\varepsilon^α)$ with $0<α<\frac{1}{2}$.
We consider a chemotaxis-Navier-Stokes system modelling cellular swimming in fluid drops where an exchange of oxygen between the drop and its environment is taken into account. This phenomenon results in an inhomogeneous Robin-type boundary condition. Moreover, the system is studied without the logistic growth of the bacteria population. We prove that in two dimensions, the system has a unique global classical solution, while the existence of a global weak solution is shown in three dimensions. In the latter case, we show that the energy is bounded uniformly in time. A key idea is to utilise a…
In this paper we study the regularity problem of a three dimensional chemotaxis-Navier-Stokes system on a periodic domain. A new regularity criterion in terms of only low modes of the oxygen concentration and the fluid velocity is obtained via a wavenumber splitting approach. The result improves many existing criteria in the literature.
We investigate the uniform regularity and vanishing viscosity limit for the incompressible chemotaxis-Navier-Stokes system in a smooth bounded domain $Ω\subset\mathbb{R}^3$. It is shown that there exists a unique strong solution of the incompressible chemotaxis-Navier-Stokes system in a finite time interval which is independent of the viscosity coefficient. Moreover, the solution is uniformly bounded in a conormal Sobolev space, which allows us to take the vanishing viscosity limit to obtain the incompressible inviscid chemotaxis-Navier-Stokes system.
In this paper, we investigate the global solvability of the chemotaxis-Navier-Stokes system on a three-dimensional moving domain of finite depth, bounded below by a rigid flat bottom and bounded above by the free surface. Completing the system with boundary conditions that match the boundary descriptions in the experiments and numerical simulations, we establish the global existence and uniqueness of solutions near a constant state $(0,\hat{c},\mathbf{0})$, where $\hat{c}$ is the saturation value of the oxygen on the free surface. To the best of our knowledge, this is the first analytical work…
We consider two dimensional Keller-Segel equations coupled with the Navier-Stokes equations modelled by Tuval et al.[32]. Assuming that the chemotactic sensitivity and oxygen consumption rate are nondecreasing and differentiable, we prove that there is no blow-up in a finite time for solutions with large initial data to chemotaxis-Navier-Stokes equations in two dimensions. In addition, temporal decays of solutions are shown, as time tends to infinity.
In this paper, we study the rates of convergence of supposedly given global bounded classical solutions to a two-species chemotaxis-Navier-Stokes system with Lotka-Volterra competitive kinetics. Except in one case where the rate of convergence for the fluid component is expressed in terms of the Poincare constant and the model parameters, all other rates of convergence are shown to be expressible only in terms of the model parameters and the underlying space dimension.
This paper is concerned with the 3-dimensional two-species chemotaxis-Navier--Stokes system with Lotka--Volterra competitive kinetics under homogeneous Neumann boundary conditions and initial conditions. Recently, in the 2-dimensional setting, global existence and stabilization of classical solutions to the above system were first established. However, the 3-dimensional case has not been studied: Because of difficulties in the Navier--Stokes system, we can not expect existence of classical solutions to the above system. The purpose of this paper is to obtain global existence of weak solutions …
In this article, we study a mathematical system which models the dynamic of the collective behaviour of oxygen-driven swimming bacteria in an aquatic fluid flowing in a two dimensional bounded domain under stochastic perturbation. This model can be seen as a stochastic version of Chemotaxis-Navier-Stokes model. We prove the existence of a unique (probabilistic) strong solution. In addition, we establish some properties of the strong solution. More precisely, we prove that the unique solution is non-negative and satisfies the mass conservation property and an energy inequality.
We consider a coupled system of Keller-Segel type equations and the incompressible Navier-Stokes equations in spatial dimension two and three. In the previous work [19], we established the existence of a weak solution of a Fokker-Plank equation in the Wasserstein space using the optimal transportation technique. Exploiting this result, we constructed solutions of Keller-Segel-Navier-Stokes equations such that the density of biological organism belongs to the absolutely continuous curves in the Wasserstein space. In this work, we refine the result on the existence of a weak solution of a Fokker…
A three-dimensional chemotaxis-Navier-Stokes system is considered. It is known that for all suitably regular initial data, a corresponding initial-boundary value problem admits at least one global weak solution which can be obtained as the pointwise limit of a sequence of solutions to appropriately regularized problems. The present paper shows that after some relaxation time, this solution enjoys further regularity properties and thereby complies with the concept of eventual energy solutions which is newly introduced here, and which inter alia requires that two quasi-dissipative inequalities a…
This paper considers the degenerate and singular chemotaxis-Navier--Stokes system with logistic term $n_t + u\cdot\nabla n =Δn^m - χ\nabla\cdot(n\nabla c) + κn -μn^2$, $x \in Ω,\ t>0$, $c_t + u\cdot\nabla c = Δc - nc$, $x \in Ω,\ t>0$, $u_t + (u\cdot\nabla)u = Δu + \nabla P + n\nablaΦ, \quad \nabla\cdot u = 0$, $x \in Ω,\ t>0$, where $Ω\subset \mathbb{R}^3$ is a bounded domain and $χ,κ\ge 0$ and $m, μ>0$. In the above system without fluid environment Jin (J. Differential Equations, 2017) showed existence and boundedness of global weak solutions. On the other hand, in the above system with $m=1…
We consider an initial-boundary value problem for the incompressible chemotaxis-Navier-Stokes equations generalizing the porous-medium-type diffusion model $ \quad n_t+u\cdot\nabla n=Δn^m-\nabla\cdot(nχ(c)\nabla c), $ $ \quad c_t+u\cdot\nabla c=Δc-nf(c), $ $ \quad u_t+κ(u\cdot\nabla)u=Δu+\nabla P+n\nablaΦ, $ $ \quad \nabla\cdot u=0, $ in a bounded convex domain $Ω\subset\mathbb{R}^3$. It is proved that if $m\geq\frac{2}{3}$, $κ\in\mathbb{R}$, $0<χ\in C^2([0,\infty))$, $0\leq f\in C^1([0,\infty))$ with $f(0)=0$ and $Φ\in W^{1,\infty}(Ω)$, then for sufficiently smooth initial data $(n_0, c_0, u_…
We analyze a diffuse interface model that describes the dynamics of incompressible two-phase flows with chemotaxis effect. The PDE system couples a Navier-Stokes equation for the fluid velocity, a convective Cahn-Hilliard equation for the phase field variable with an advection-diffusion-reaction equation for the nutrient density. For the system with a singular potential, we prove the existence of global weak solutions in both two and three dimensions. Besides, in the two dimensional case, we establish a continuous dependence result that implies the uniqueness of global weak solutions. The sing…
The chemotaxis-Navier-Stokes system linking the chemotaxis equations \[ n_t + u\cdot\nabla n = Δn - \nabla \cdot (nχ(c)\nabla c) \] and \[ c_t + u\cdot\nabla c = Δc-nf(c) \] to the incompressible Navier-Stokes equations, \[ u_t + (u\cdot\nabla)u = Δu +\nabla P + n \nabla Φ, \qquad \nabla \cdot u = 0, \] is considered under homogeneous boundary conditions of Neumann type for $n$ and $c$, and of Dirichlet type for $u$, in a bounded convex domain $Ω\subset R^3$ with smooth boundary, where $Φ\in W^{1,\infty}(Ω)$, and where $f\in C^1([0,\infty))$ and $χ\in C^2([0,\infty))$ are nonnegative with $f(0…
In this work we consider the Keller-Segel system coupled with Navier-Stokes equations in $\mathbb{R}^{N}$ for $N\geq2$. We prove the global well-posedness with small initial data in Besov-Morrey spaces. Our initial data class extends previous ones found in the literature such as that obtained by Kozono-Miura-Sugiyama (J. Funct. Anal. 2016). It allows to consider initial cell density and fluid velocity concentrated on smooth curves or at points depending on the spatial dimension. Self-similar solutions are obtained depending on the homogeneity of the initial data and considering the case of che…
This paper investigates an incompressible chemotaxis-Navier-Stokes system with slow $p$-Laplacian diffusion \begin{eqnarray} \left\{\begin{array}{lll} n_t+u\cdot\nabla n=\nabla\cdot(|\nabla n|^{p-2}\nabla n)-\nabla\cdot(nχ(c)\nabla c),& x\inΩ,\ t>0, c_t+u\cdot\nabla c=Δc-nf(c),& x\inΩ,\ t>0, u_t+(u\cdot\nabla) u=Δu+\nabla P+n\nablaΦ,& x\inΩ,\ t>0, \nabla\cdot u=0,& x\inΩ,\ t>0 \end{array}\right. \end{eqnarray} under homogeneous boundary conditions of Neumann type for $n$ and $c$, and of Dirichlet type for $u$ in a bounded convex domain $Ω\subset \mathbb{R}^3$ with smooth boundary. Here, $Φ\in …
We show that the incompressible 3D Navier-Stokes system in a C 1;1 bounded domain or a bounded convex domain with a non penetration condition u = 0 at the boundary @ together with a time-dependent Robin boundary condition of the type curl u = (t)u on @ admits a solution with enough regularity provided the initial condition is small enough in an appropriate functional space.
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 r…
We prove two new functional inequalities of the forms\[ \int_G φ(ψ- \overlineψ) \leq \frac{1}{a}\int_G ψ\ln \left(\frac{\;ψ\;}{ \overlineψ}\right) + \frac{a}{4β_0} \left\{ \int_G ψ\right\}\int_G|\nabla φ|^2 \] and \[ \int_G ψ\ln \left(\frac{\;ψ\;}{ \overlineψ}\right) \leq \frac{1}{β_0}\left\{ \int_G ψ\right\}\int_G |\nabla \ln(ψ)|^2 \] for any finitely connected, bounded $C^2$-domain $G \subseteq \mathbb{R}^2$, a constant $β_0 > 0$, any $a > 0$ and sufficiently regular functions $φ$, $ψ$. We then illustrate their usefulness by proving long time stabilization and eventual smoothness properties …
We consider initial-boundary value problems for the $κ$-dependent family of chemotaxis-(Navier--)Stokes systems \begin{align*} \left\{ \begin{array}{r@{\,}c@{\,}c@{\ }l@{\quad}l@{\quad}l@{\,}c} n_{t}&+&u\cdot\!\nabla n&=Δn-\nabla\!\cdot(n\nabla c),\ &x\inΩ,& t>0,\\ c_{t}&+&u\cdot\!\nabla c&=Δc-cn,\ &x\inΩ,& t>0,\\ u_{t}&+&κ(u\cdot\nabla)u&=Δu+\nabla P+n\nablaϕ,\ &x\inΩ,& t>0,\\ &&\nabla\cdot u&=0,\ &x\inΩ,& t>0, \end{array}\right. \end{align*} in a bounded domain $Ω\subset\mathbb{R}^3$ with smooth boundary and given potential function $ϕ\in C^{1+β}(\overlineΩ)$ for some $β>0$. It is known that…
In this work, we study a chemotaxis-Navier-Stokes model in a two-dimensional setting as below, \begin{eqnarray} \left\{ \begin{array}{llll} \displaystyle n_{t}+\mathbf{u}\cdot\nabla n=Δn-\nabla \cdot(n\nabla c)+f(n), &&x\inΩ,\,t>0,\\ \displaystyle c_{t}+\mathbf{u}\cdot\nabla c=Δc - c+ n, &&x\inΩ,\,t>0,\\ \displaystyle \mathbf{u}_{t}+κ(\mathbf{u}\cdot\nabla)\mathbf{u}=Δ\mathbf{u} +\nabla P+ n\nablaϕ, &&x\inΩ,\,t>0,\\ \displaystyle \nabla\cdot\mathbf{u}=0,&&x\inΩ,\,t>0.\\ \end{array} \right. \end{eqnarray} Motivated by a recent work due to Winkler, we aim at investigating generalized solvability…
This paper is concerned with the following quasilinear chemotaxis--Navier--Stokes system with nonlinear diffusion and rotation $$ \left\{ \begin{array}{l} n_t+u\cdot\nabla n=Δn^m-\nabla\cdot(nS(x,n,c)\cdot\nabla c),\quad x\in Ω, t>0, c_t+u\cdot\nabla c=Δc-nc,\quad x\in Ω, t>0,\\ u_t+κ(u \cdot \nabla)u+\nabla P=Δu+n\nabla ϕ,\quad x\in Ω, t>0,\\ \nabla\cdot u=0,\quad x\in Ω, t>0 \end{array}\right.\eqno(CNF) $$ is considered under the no-flux boundary conditions for $n, c$ and the Dirichlet boundary condition for $u$ in a three-dimensional convex domain $Ω\subseteq \mathbb{R}^3$ with smooth bound…
Global existence and boundedness of classical solutions of the chemotaxis--consumption system \begin{align*} n_t &= Δn - \nabla \cdot (n \nabla c), \\ 0 &= Δc - nc, \end{align*} under no-flux boundary conditions for $n$ and Robin-type boundary conditions \[ \partial_ν c = (γ-c) g \] for $c$ (with $γ>0$ and $C^{1+β}(\partialΩ) \ni g > 0$ for some $β\in(0,1)$) are established in bounded domains $Ω\subset\mathbb{R}^{N}$, $N\ge 1$. Under a smallness condition on $γ$, moreover, we show convergence to the stationary solution.
The chemotaxis--Navier--Stokes system \begin{equation*}\label{0.1} \left\{\begin{array}{ll} n_t+u\cdot \nabla n=\triangle n-χ\nabla\cdotp \left(\displaystyle\frac n {c}\nabla c\right)+n(r-μn), c_t+u\cdot \nabla c=\triangle c-nc, u_t+ (u\cdot \nabla) u=Δu+\nabla P+n\nablaϕ, \nabla\cdot u=0, \end{array}\right. \end{equation*} is considered in a bounded smooth domain $Ω\subset \mathbb{R}^2$, where $ϕ\in W^{1,\infty}(Ω)$, $χ>0$, $r\in \mathbb{R}$ and $μ> 0$ are given parameters. It is shown that there exists a value $μ_*(Ω,χ, r)\geq 0$ such that whenever $ μ>μ_*(Ω,χ, r)$, the global-in-time classi…
We study the Cauchy problem for the chemotaxis Navier-Stokes equations and the Keller-Segel-Navier-Stokes system. Local-in-time and global-in-time solutions satisfying fundamental properties such as mass conservation and nonnegativity preservation are constructed for low regularity data in $2$ and higher dimensions under suitable conditions. Our initial data classes involve a new scale of function space, that is $\Y(\rn)$ which collects divergence of vector-fields with components in the square Campanato space $\mathscr{L}_{2,N-2}(\rn)$, $N>2$ (and can be identified with the homogeneous Besov s…
We consider the coupled chemotaxis Navier-Stokes model with logistic source terms \[ n_t + u\cdot \nabla n = Δn - χ\nabla \cdot (n \nabla c) + κn - μn^2\] \[ c_t + u\cdot \nabla c = Δc - nc\] \[ u_t + (u\cdot \nabla)u = Δu +\nabla P + n\nabla Φ+ f, \quad\qquad \nabla \cdot u=0 \] in a bounded, smooth domain $Ω\subset \mathbb{R}^3$ under homogeneous Neumann boundary conditions for $n$ and $c$ and homogeneous Dirichlet boundary conditions for $u$ and with given functions $f\in L^\infty(Ω\times(0,\infty))$ satisfying certain decay conditions and $Φ\in C^{1+β}(\barΩ)$ for some $β\in(0,1)$. We cons…
This paper considers the chemotaxis-Navier--Stokes system with nonlinear diffusion and logistic-type degradation term \begin{align*} \begin{cases} n_t + u\cdot\nabla n = \nabla \cdot(D(n)\nabla n) - \nabla\cdot(n χ(c) \nabla c) + κn - μn^α, & x\in Ω,\ t>0, \\ c_t + u\cdot\nabla c = Δc - nf(c), & x \in Ω,\ t>0, \\ u_t + (u\cdot\nabla)u = Δu + \nabla P + n\nablaΦ+ g, \ \nabla\cdot u = 0, & x \in Ω,\ t>0, \end{cases} \end{align*} where $Ω\subset \mathbb{R}^3$ is a bounded smooth domain; $D \ge 0$ is a given smooth function such that $D_1 s^{m-1} \le D(s) \le D_2 s^{m-1}$ for all $s\ge 0$ with som…
The coupled chemotaxis fluid system \begin{equation} \left\{ \begin{array}{llc} \displaystyle n_t=Δn-\nabla\cdot(nS(x,n,c)\cdot\nabla c)-u\cdot\nabla n, &(x,t)\in Ω\times (0,T),\\ c_t=Δc-nc-u\cdot\nabla c , &(x,t)\inΩ\times (0,T),\\ u_t=Δu-κ(u\cdot\nabla)u+\nabla P+n\nablaϕ, &(x,t)\inΩ\times (0,T),\\ \nabla\cdot u=0,&(x,t)\inΩ\times (0,T), \end{array} \right.(\star) \end{equation} is considered under the no-flux boundary conditions for $n,c$ and the Dirichlet boundary condition for $u$ on a bounded smooth domain $Ω\subset\mathbb{R}^N$ ($N=2,3$), $κ=0,1$. We assume that $S(x,n,c)$ is a matrix-v…
We study the chemotaxis-Navier-Stokes system \[\left\{\; \begin{aligned} n_t + u\cdot\nabla n &=Δn - \nabla\cdot (nS(x,n,c)\nabla c), &&x\inΩ, t > 0, \\ c_t + u\cdot\nabla c &=Δc - n f(c), && x\in Ω, t > 0, \\ u_t + (u\cdot\nabla) u &=Δu + \nabla P + n \nabla ϕ, \;\;\;\; \nabla\cdot u = 0, && x\inΩ, t > 0 \end{aligned}\right.\tag{$\star$} \] with no-flux boundary conditions for $n$, $c$ in a bounded, convex domain $Ω\subseteq\mathbb{R}^2$ with a smooth boundary, which is motivated by recent modeling approaches from biology for aerobic bacteria suspended in a sessile water drop. We further do n…
The existence and uniqueness of the analytic solutions to the nonlinear Prandtl equations with Robin boundary condition on a half space are proved, based on an application of abstract Cauchy-Kowalewski theorem. These equations arise in the inviscid limit of incompressible Navier-Stokes equations with Navier-slip boundary condition in which the slip length is square root of viscosity, as formally derived in [26].
Regularity Propagation of Global Weak Solutions to a Navier-Stokes-Cahn-Hilliard System for Incompressible Two-phase Flows with Chemotaxis and Active Transport
In this paper, we investigate a chemotaxis-fluid system involving both the effect of potential force on cells and the effect of chemotactic force on fluid: \begin{equation*} \left\{ \begin{split} \partial_t n + \mathbf{u}\cdot\nabla n & = Δn - \nabla\cdot\left(χ(c)n\nabla c\right) + \nabla\cdot(n\nablaϕ), \\ \partial_t c + \mathbf{u}\cdot\nabla c &= Δc - nf(c), \\ \partial_t\mathbf{u} + κ(\mathbf{u}\cdot\nabla)\mathbf{u} + \nabla P & = Δ\mathbf{u} - n\nablaϕ+ χ(c)n\nabla c, \\ \nabla\cdot\mathbf{u} &= 0 \end{split} \right. \end{equation*} in $\mathbb{R}^d\times(0,T)\, (d=2,3)$. One of the nove…
We consider a model of congestion dynamics with chemotaxis, where the density of cells follows the chemical signal it generates, while observing an incompressibility constraint. We show that when the chemical diffuses slowly and attracts the cells strongly, then the dynamics of the congested cells is well approximated by a surface-tension driven free boundary problem. More precisely, we show that in this limit the density of cell converges to the characteristic function of a set whose evolution is described by a Hele-Shaw free boundary problem with surface tension. Our problem is set in a boun…
