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In this paper, we study a nutrient-taxis model with porous medium slow diffusion \begin{align*} \left\{ \begin{aligned} &u_t=\Delta u^m-\chi\nabla\cdot(u\nabla v)+\xi uv-\rho u, \\ &v_t-\Delta v=-vu+\mu v(1-v), \end{aligned}\right.…
In a smoothly bounded domain $\Omega \subset \mathbb{R}^N$ $(N\in \mathbb{N})$, a no-flux initial-boundary value problem for the degenerate chemotaxis system with volume-filling effects, \begin{align*} u_t = \nabla \cdot (D(u,v) \nabla u -…
This paper studies the dynamical behavior of classical solutions to a hyperbolic system of balance laws, derived from a chemotaxis model with logarithmic sensitivity, subject to time-dependent boundary conditions. It is shown that under…
In recent years, a lot of attention has been drawn to the question of whether logistic kinetics is sufficient to enforce the global existence of classical solutions or to prevent finite-time blow-up in various chemotaxis models. The current…
We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the isentropic compressible Navier-Stokes equations in three spatial dimensions with smooth initial data which are of small energy but…
This paper deals with the following parabolic-elliptic chemotaxis system with singular sensitivity and logistic source, \begin{equation} \begin{cases} u_t=\Delta u-\chi\nabla\cdot (\frac{u}{v} \nabla v)+u(a(t,x)-b(t,x) u), & x\in \Omega,\cr…
The micropolar equations are a useful generalization of the classical Navier-Stokes model for fluids with micro-structure. We prove the existence of global and strong solutions to these equations in cylindrical domains in $\mathbb{R}^3$. We…
While much literature on chemotaxis systems focuses on bounded domains, this paper emphasizes the global existence of classical solutions for three primary chemotaxis systems with a logistic source on $\mathbb{R}^n$. We present a unified…
Consider the coupled Keller-Segel-Navier-Stokes or the chemotaxis-consumption-Navier-Stokes system in bounded Lipschitz domains for general coupling terms which, e.g., include buoyancy forces. It is shown that these systems admit local…
In this paper, we investigate the existence of a unique global smooth solution to the three-dimensional incompressible Navier-Stokes equations and provide a concise proof. We establish a new global well-posedness result that allows the…
We consider classical solutions to the chemotaxis system with logistic source $f(u) := au-\mu u^2$ under nonlinear Neumann boundary condition $\frac{\partial u}{ \partial \nu } = |u|^{p}$ with $p>1$ in a smooth convex bounded domain $\Omega…
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…
We consider the potentially degenerate haptotaxis system \begin{equation*} \left\{ \begin{aligned} u_t &= \nabla \cdot (\mathbb{D} \nabla u + u \nabla \cdot \mathbb{D}) - \chi \nabla \cdot (u\mathbb{D}\nabla w) + \mu u(1-u^{r- 1}), \\ w_t…
We consider an initial-boundary value problem for the incompressible four-component Keller-Segel-Navier-Stokes system with rotational flux $$\left\{\begin{array}{l} n_t+u\cdot\nabla n=\Delta n-\nabla\cdot(nS(x,n,c)\nabla c)-nm,\quad x\in…
This paper is concerned with the 3-dimensional two-species chemotaxis-Navier--Stokes system with Lotka--Volterra competitive kinetics under homogeneous Neumann boundary conditions and initial conditions. Recently, in the 2-dimensional…
This paper deals with the long-term behavior of positive solutions for the following parabolic-elliptic chemotaxis competition system with weak singular sensitivity and logistic source \begin{equation} \label{abstract-eq} \begin{cases}…
The coupled quasilinear Keller-Segel-Navier-Stokes system $$ \left\{ \begin{array}{l} n_t+u\cdot\nabla n=\Delta n-\nabla\cdot(nS(x,n,c)\nabla c),\quad x\in \Omega, t>0, c_t+u\cdot\nabla c=\Delta c-c+n,\quad x\in \Omega, t>0, u_t+\kappa(u…
In this paper, we consider the three-dimensional inhomogeneous Navier-Stokes equations with density-dependent viscosity in presence of vacuum over bounded domains. Global-in-time unique strong solution is proved to exist when $\|\nabla…
We derive a class of Navier--Stokes--Cahn--Hilliard systems that models two-phase flows with mass transfer coupled to the process of chemotaxis. These thermodynamically consistent models can be seen as the natural Navier--Stokes analogues…
We prove the global existence and uniqueness of the classical (weak) solution for the 2D or 3D compressible Navier-Stokes equations with a density-dependent viscosity coefficient ($\lambda=\lambda(\rho)$). Initial data and solutions are…