Global very weak solutions to a chemotaxis-fluid system with nonlinear diffusion

We consider the chemotaxis-fluid system \begin{align}\label{star}\tag{$\diamondsuit$} \left\{ \begin{array}{r@{\,}c@{\,}c@{\ }l@{\quad}l@{\quad}l@{\,}c} n_{t}&+&u\cdot\!\nabla n&=\Delta n^m-\nabla\!\cdot(n\nabla c),\ &x\in\Omega,& t>0,\\ c_{t}&+&u\cdot\!\nabla c&=\Delta c-c+n,\ &x\in\Omega,& t>0,\\ u_{t}&+&(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{align} in a bounded domain $\Omega\subset\mathbb{R}^3$ with smooth boundary and $m>1$. Assuming $m>\frac{4}{3}$ and sufficiently regular nonnegative initial data, we ensure the existence of global solutions to the no-flux-Dirichlet boundary value problem for \eqref{star} under a suitable notion of very weak solvability, which in different variations has been utilized in the literature before. Comparing this with known results for the fluid-free setting of \eqref{star} the condition appears to be optimal with respect to global existence. In case of the stronger assumption $m>\frac{5}{3}$ we moreover establish the existence of at least one global weak solution in the standard sense. In our analysis we investigate a functional of the form $\int_{\Omega}\! n^{m-1}+\int_{\Omega}\! c^2$ to obtain a spatio-temporal $L^2$ estimate on $\nabla n^{m-1}$, which will be the starting point in deriving a series of compactness properties for a suitably regularized version of \eqref{star}. As the regularity information obtainable from these compactness results vary depending on the size of $m$, we will find that taking $m>\frac{5}{3}$ will yield sufficient regularity to pass to the limit in the integrals appearing in the weak formulation, while for $m>\frac{4}{3}$ we have to rely on milder regularity requirements making only very weak solutions attainable.Comment: 24 pages, closed a gap underlying the proof of Remark 2.4 ii), reformulated some parts accordingly and fixed typo

How strongly does diffusion or logistic-type degradation affect existence of global weak solutions in a chemotaxis-Navier--Stokes system?

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 \chi(c) \nabla c) + \kappa n - \mu n^\alpha, & x\in \Omega,\ t>0, \\ c_t + u\cdot\nabla c = \Delta c - nf(c), & x \in \Omega,\ t>0, \\ u_t + (u\cdot\nabla)u = \Delta u + \nabla P + n\nabla\Phi + g, \ \nabla\cdot u = 0, & x \in \Omega,\ t>0, \end{cases} \end{align*} where $\Omega\subset \mathbb{R}^3$ 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 some $D_2 \ge D_1 > 0$ and some $m > 0$; $\chi,f$ are given functions satisfying some conditions; $\kappa \in \mathbb{R},\mu \ge0,\alpha>1$ are constants. This paper shows existence of global weak solutions to the above system under the condition that \begin{align*} m >\frac{2}{3},\quad \mu \ge 0 \quad \mbox{and}\quad \alpha >1 \end{align*} hold, or that \begin{align*} m> 0, \quad \mu>0 \quad \mbox{and} \quad \alpha > \frac{4}{3} \end{align*} hold. This result asserts that `strong' diffusion effect or `strong' logistic damping derives existence of global weak solutions even though the other effect is `weak', and can include previous works.Comment: 29 page

Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with nonlinear diffusion

We consider an initial-boundary value problem for the incompressible chemotaxis-Navier-Stokes equations generalizing the porous-medium-type diffusion model $\quad n_t+u\cdot\nabla n=\Delta n^m-\nabla\cdot(n\chi(c)\nabla c),$ $\quad c_t+u\cdot\nabla c=\Delta c-nf(c),$ $\quad u_t+\kappa(u\cdot\nabla)u=\Delta u+\nabla P+n\nabla\Phi,$ $\quad \nabla\cdot u=0,$ in a bounded convex domain $\Omega\subset\mathbb{R}^3$. It is proved that if $m\geq\frac{2}{3}$, $\kappa\in\mathbb{R}$, $0<\chi\in C^2([0,\infty))$, $0\leq f\in C^1([0,\infty))$ with $f(0)=0$ and $\Phi\in W^{1,\infty}(\Omega)$, then for sufficiently smooth initial data $(n_0, c_0, u_0)$ the model possesses at least one global weak solution

Global solvability of chemotaxis-fluid systems with nonlinear diffusion and matrix-valued sensitivities in three dimensions

In this work we extend a recent result to chemotaxis fluid systems which include matrix-valued sensitivity functions $S(x,n,c):\Omega\times[0,\infty)^2\to\mathbb{R}^{3\times3}$ in addition to the porous medium type diffusion, which were discussed in the previous work. Namely, we will consider the system \begin{align*} \left\{ \begin{array}{r@{\,}c@{\,}c@{\ }l@{\quad}l@{\quad}l@{\,}c} n_{t}&+&u\cdot\!\nabla n&=\Delta n^m-\nabla\!\cdot(nS(x,n,c)\nabla c),\ &x\in\Omega,& t>0,\\ c_{t}&+&u\cdot\!\nabla c&=\Delta c-c+n,\ &x\in\Omega,& t>0,\\ u_{t}&+&(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{align*} in a bounded domain $\Omega\subset\mathbb{R}^3$ with smooth boundary. Assuming that $m\geq1$, $\alpha\geq0$ satisfy $m+\alpha>\frac43$, that the matrix-valued function $S(x,n,c):\Omega\times[0,\infty)^2\to\mathbb{R}^{3\times3}$ satisfies $|S(x,n,c)|\leq\frac{S_0}{(1+n)^{\alpha}}$ for some $S_0>0$ and suitably regular nonnegative initial data, we show that the corresponding no-flux-Dirichlet boundary value problem emits at least one global very weak solution. Upon comparison with results for the fluid-free system this condition appears to be optimal. Moreover, imposing a stronger condition for the exponents $m$ and $\alpha$, i.e. $m+2\alpha>\frac{5}{3}$, we will establish the existence of at least one global weak solution in the standard sense.Comment: 23 page

Global existence to a 3D3D chemotaxis-Navier-stokes system with nonlinear diffusion and rotation

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=\Delta n^m-\nabla\cdot(nS(x,n,c)\cdot\nabla c),\quad x\in \Omega, t>0, c_t+u\cdot\nabla c=\Delta c-nc,\quad x\in \Omega, t>0,\\ u_t+\kappa(u \cdot \nabla)u+\nabla P=\Delta u+n\nabla \phi ,\quad x\in \Omega, t>0,\\ \nabla\cdot u=0,\quad x\in \Omega, 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 $\Omega\subseteq \mathbb{R}^3$ with smooth boundary, which describes the motion of oxygen-driven bacteria in a fluid. Here % $\Omega\subseteq \mathbb{R}^3$ is a , $\kappa\in \mathbb{R}$ and $S$ denotes the strength of nonlinear fluid convection and a given tensor-valued function, respectively. Assume $m>\frac{10}{9}$ and $S$ fulfills $|S(x,n,c)| \leq S_0(c)$ for all $(x,n,c)\in \bar{\Omega} \times [0, \infty)\times[0, \infty)$ with $S_0(c)$ nondecreasing on $[0,\infty)$, then for any reasonably regular initial data, the corresponding initial-boundary problem $(CNF)$ admits at least one global weak solution

Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system

The chemotaxis-Navier-Stokes system linking the chemotaxis equations $$n_t + u\cdot\nabla n = \Delta n - \nabla \cdot (n\chi(c)\nabla c)$$ and $$c_t + u\cdot\nabla c = \Delta c-nf(c)$$ to the incompressible Navier-Stokes equations, $$u_t + (u\cdot\nabla)u = \Delta u +\nabla P + n \nabla \Phi, \qquad \nabla \cdot u = 0,$$ is considered under homogeneous boundary conditions of Neumann type for $n$ and $c$, and of Dirichlet type for $u$, in a bounded convex domain $\Omega\subset R^3$ with smooth boundary, where $\Phi\in W^{1,\infty}(\Omega)$, and where $f\in C^1([0,\infty))$ and $\chi\in C^2([0,\infty))$ are nonnegative with $f(0)=0$. Problems of this type have been used to describe the mutual interaction of populations of swimming aerobic bacteria with the surrounding fluid. Up to now, however, global existence results seem to be available only for certain simplified variants such as e.g.the two-dimensional analogue, or the associated chemotaxis-Stokes system obtained on neglecting the nonlinear convective term in the fluid equation. The present work gives an affirmative answer to the question of global solvability in the following sense: Under mild assumptions on the initial data, and under modest structural assumptions on $f$ and $\chi$, inter alia allowing for the prototypical case when $$f(s)=s \quad {for all} s\ge 0 \qquad {and} \qquad \chi \equiv const.,$$ the corresponding initial-boundary value problem is shown to possess a globally defined weak solution

Singular sensitivity in a Keller-Segel-fluid system

In bounded smooth domains $\Omega\subset\mathbb{R}^N$, $N\in\{2,3\}$, considering the chemotaxis--fluid system \begin{cases} \begin{split} & n_t + u\cdot \nabla n &= \Delta n - \chi \nabla \cdot(\frac{n}{c}\nabla c) &\\ & c_t + u\cdot \nabla c &= \Delta c - c + n &\\ & u_t + \kappa (u\cdot \nabla) u &= \Delta u + \nabla P + n\nabla \Phi & \end{split}\end{cases} with singular sensitivity, we prove global existence of classical solutions for given $\Phi\in C^2(\bar{\Omega})$, for $\kappa=0$ (Stokes-fluid) if $N=3$ and $\kappa\in\{0,1\}$ (Stokes- or Navier--Stokes fluid) if $N=2$ and under the condition that \[ 0<\chi<\sqrt{\frac{2}{N}}. \

Global classical solutions in chemotaxis(-Navier)-Stokes system with rotational flux term

The coupled chemotaxis fluid system \begin{equation} \left\{ \begin{array}{llc} \displaystyle n_t=\Delta n-\nabla\cdot(nS(x,n,c)\cdot\nabla c)-u\cdot\nabla n, &(x,t)\in \Omega\times (0,T),\\ c_t=\Delta c-nc-u\cdot\nabla c , &(x,t)\in\Omega\times (0,T),\\ u_t=\Delta u-\kappa(u\cdot\nabla)u+\nabla P+n\nabla\phi , &(x,t)\in\Omega\times (0,T),\\ \nabla\cdot u=0,&(x,t)\in\Omega\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 $\Omega\subset\mathbb{R}^N$ ($N=2,3$), $\kappa=0,1$. We assume that $S(x,n,c)$ is a matrix-valued sensitivity under a mild assumption such that $|S(x,n,c)|<S_0(c_0)$ with some non-decreasing function $S_0\in C^2((0,\infty))$. It contrasts the related scalar sensitivity case that $(\star)$ does not possess the natural {\em gradient-like} functional structure. Associated estimates based on the natural functional seem no longer available. In the present work, a global classical solution is constructed under a smallness assumption on $\|c_0\|_{L^\infty(\Omega)}$ and moreover we obtain boundedness and large time convergence for the solution, meaning that small initial concentration of chemical forces stabilization

Existence of global solutions for a Keller-Segel-fluid equations with nonlinear diffusion

We consider a coupled system consisting of the Navier-Stokes equations and a porous medium type of Keller-Segel system that model the motion of swimming bacteria living in fluid and consuming oxygen. We establish the global-in-time existence of weak solutions for the Cauchy problem of the system in dimension three. In addition, if the Stokes system, instead Navier-Stokes system, is considered for the fluid equation, we prove that bounded weak solutions exist globally in time.Comment: 24page

Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity

We consider a chemotaxis-fluid system involving nonlinear cell diffusion of porous medium type, signal consumption by cells, and rather general, possibly matrix-valued, chemotactic sensitivities. It is shown that if the corresponding diffusion exponent $m$ satisfies $m>7/6$, then for all reasonably regaular initial data an associated initial-boundary value problem in smoothly bounded three-dimensional domains possesses a globally defined weak solution which is bounded. Under a mild additional assumption on the signal consumption rate, it is moreover shown that any nontrivial of these solutions stabilizes toward a spatially homogeneous equilibrium in the large time limit