A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

The fully parabolic Keller--Segel system is coupled to the incompressible Navier--Stokes equations through transport and buoyancy. It is shown that when posed with no-flux/no-flux/Dirichlet boundar...

Small-Mass Solutions in the Two-Dimensional Keller--Segel System Coupled to the Navier--Stokes Equations

This paper focuses on the following Keller–Segel–Navier–Stokes system with rotational flux: in a bounded domain $$\Omega \subset {\mathbb {R}}^3$$
 with a smooth boundary, where $$\kappa \in {\mathbb {R}}$$
 is a given constant, $$\phi \in W^{1,\infty }(\Omega )$$
 , $$|S(x,n,c)|\le C_S(1+n)^{-\alpha }$$
 , and the parameter $$\alpha \ge 0$$
 . If $$\alpha >\frac{1}{3}$$
 , then, for all reasonable regular initial data, a corresponding initial-boundary value problem for (KSNF) possesses a globally defined weak solution. This result improves upon the result of Wang (Math Models Methods Appl Sci 27(14):2745–2780, 2017), in which the global very weak solution for the system (KSNF) is obtained. In comparison with the result of the corresponding fluid-free system, the optimal condition on the parameter $$\alpha $$
 for global (weak) existence is established. Our proofs rely on a variant of the natural gradient-like energy functional.

An optimal result for global existence in a three-dimensional Keller–Segel–Navier–Stokes system involving tensor-valued sensitivity with saturation

The coupled quasilinear Keller-Segel-Navier-Stokes system is considered under Neumann boundary conditions for $n$ and $c$ and no-slip boundary conditions for $u$ in three-dimensional bounded domains $\Omega\subseteq \mathbb{R}^3$ with smooth boundary, where $m>0,\kappa\in \mathbb{R}$ are given constants, $\phi\in W^{1,\infty}(\Omega)$. If $ m> 2$, then for all reasonably regular initial data, a corresponding initial-boundary value problem for $(KSNF)$ possesses a globally defined weak solution.

Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion

Global existence and asymptotic behavior in a two-species chemotaxis system with logistic source

An optimal result for global existence and boundedness in a three-dimensional Keller-Segel-Stokes system with nonlinear diffusion

This paper investigates the following quasilinear Keller-Segel-Navier-Stokes system $$\left\{ \begin{array}{l} n_t+u\cdot\nabla n=\Delta n^m-\nabla\cdot(n\nabla c),\quad x\in \Omega, t>0, \\ c_t+u\cdot\nabla c=\Delta c-c+n,\quad x\in \Omega, t>0,\\ u_t+\nabla P=\Delta u+n\nabla \phi,\quad x\in \Omega, t>0,\\ \nabla\cdot u=0, \quad x\in \Omega, t>0 \end{array}\right.$$ under homogeneous boundary conditions of Neumann type for $n$ and $c$, and of Dirichlet type for $u$ in a three-dimensional bounded domains $\Omega\subseteq \mathbb{R}^3$ with smooth boundary, where $\phi\in W^{1,\infty}(\Omega),m>0$. It is proved that if $m>\frac{4}{3}$, then for any sufficiently regular nonnegative initial data there exists at least one global boundedness solution for system $(KSF)$, which in view of the known results for the fluid-free system mentioned below (see Introduction) is an optimal restriction on $m$.

Jiashan Zheng

Papers

An optimal result for global existence and boundedness in a three-dimensional Keller-Segel-Stokes system with nonlinear diffusion

An optimal result for global existence and boundedness in a three-dimensional Keller-Segel-Stokes system with nonlinear diffusion