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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 ,…

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, \[…

This paper is concerned with the two-dimensional chemotaxis-fluid model \begin{equation*} \begin{cases} n_t+u\cdot\nabla n=\Delta (n\phi(v))+\mu n(1-n),\\ v_t+u\cdot\nabla v=\Delta v-nv,\\ u_t+ \kappa (u\cdot\nabla) u=\Delta…

Assuming that $0<\chi<\sqrt{\frac{2}n}$, $\kappa\ge 0$ and $\mu>\frac{n-2}{n}$, we prove global existence of classical solutions to a chemotaxis system slightly generalizing \[ \begin{split} u_t &= \Delta u - \chi \nabla\cdot ( \frac{u}{v}…

We study the chemotaxis-Navier-Stokes system \[\left\{\; \begin{aligned} n_t + u\cdot\nabla n &=\Delta n - \nabla\cdot (nS(x,n,c)\nabla c), &&x\in\Omega, t > 0, \\ c_t + u\cdot\nabla c &=\Delta c - n f(c), && x\in \Omega, t > 0, \\ u_t +…

This paper studies the following chemotaxis-fluid system in a two-dimensional bounded domain $\Omega$: \begin{equation*} \begin{cases} n_t + u \cdot \nabla n &= \Delta n - \chi \nabla \cdot \left (n \frac{\nabla c}{c^k} \right ) + r n -…

This paper deals with a boundary-value problem for a coupled chemotaxis-Navier-Stokes system involving tensor-valued sensitivity with saturation $$\left\{ \begin{array}{l} n_t+u\cdot\nabla n=\Delta n-\nabla\cdot(nS(x,n,c)\nabla c),\quad…

In this paper, the three-dimensional chemotaxis-stokes system \begin{eqnarray*} \left\{\begin{array}{lll} \medskip n_{t}+u\cdot\nabla n=\Delta n^m-\nabla\cdot(n S(x,n,c)\cdot\nabla c),&x\in\Omega,\ \ t>0, \medskip c_t+u\cdot\nabla c=\Delta…

We are concerned with the Keller--Segel--Navier--Stokes system \begin{equation*} \left\{ \begin{array}{ll} \rho_t+u\cdot\nabla\rho=\Delta\rho-\nabla\cdot(\rho \mathcal{S}(x,\rho,c)\nabla c)-\rho m, &\!\! (x,t)\in \Omega\times (0,T), \\…

In this paper we study the zero-flux chemotaxis-system \begin{equation*} \begin{cases} u_t=\Delta u -\chi \nabla \cdot (\frac{u}{v} \nabla v) \\ v_t=\Delta v-f(u)v \end{cases} \end{equation*} in a smooth and bounded domain $\Omega$ of…

The chemotaxis--Navier--Stokes system \begin{equation*}\label{0.1} \left\{\begin{array}{ll} n_t+u\cdot \nabla n=\triangle n-\chi\nabla\cdotp \left(\displaystyle\frac n {c}\nabla c\right)+n(r-\mu n), c_t+u\cdot \nabla c=\triangle c-nc, u_t+…

This paper considers the degenerate and singular chemotaxis-Navier--Stokes system with logistic term $n_t + u\cdot\nabla n =\Delta n^m - \chi\nabla\cdot(n\nabla c) + \kappa n -\mu n^2$, $x \in \Omega,\ t>0$, $c_t + u\cdot\nabla c = \Delta c…

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(n\chi(c)\nabla c),&…

This paper is concerned with the singular chemotaxis-fluid system with indirect nutrient consumption: $ n_{t}+u\cdot\nabla n=\Delta n-\nabla\cdot(n S(x,n,v)\cdot \nabla v);\ v_{t}+u\cdot\nabla v=\Delta v-vw;\ w_{t}+u\cdot\nabla w=\Delta…

A class of chemotaxis-Stokes systems generalizing the prototype \[\left\{ \begin{array}{rcl} n_t + u\cdot\nabla n &=& \nabla \cdot \big(n^{m-1}\nabla n\big) - \nabla \cdot \big(n\nabla c\big), c_t + u\cdot\nabla c &=& \Delta c-nc, u_t…

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…

We study the chemotaxis-fluid system \begin{align*} \left\{ \begin{array}{r@{\,}c@{\,}c@{\ }l@{\quad}l@{\quad}l@{\,}c} n_{t}&+&u\cdot\!\nabla n&=\Delta n-\nabla\!\cdot(\frac{n}{c}\nabla c),\ &x\in\Omega,& t>0, c_{t}&+&u\cdot\!\nabla…

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…

We consider the chemotaxis-Navier-Stokes system with generalized fluid dissipation in $\mathbb{R}^3$: \begin{eqnarray*} \begin{cases} \partial_t n+u\cdot \nabla n=\Delta n- \nabla \cdot (\chi(c)n \nabla c),\\ \partial_t c+u \cdot \nabla…

We consider the chemotaxis system \begin{eqnarray*} \begin{cases} \begin{array}{lll} \medskip u_t =\Delta u^m - \nabla(\frac{u}{v}\nabla v),&{} x\in\Omega,\ t>0, \medskip v_t =\Delta v -uv,&{}x\in\Omega,\ t>0, \medskip \frac{\partial…