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

This paper deals with an initial-boundary value problem for the system

 $$\left\{ \begin{array}{llll} n_t + u\cdot\nabla n &=& \Delta n -\nabla \cdot (n\chi(c)\nabla c), \quad\quad & x\in\Omega, \, t > 0,\\ c_t + u\cdot\nabla c &=& \Delta c-nf(c), \quad\quad & x\in\Omega, \, t > 0,\\
 u_t + \kappa (u\cdot \nabla) u &=& \Delta u + \nabla P + n \nabla\phi, \qquad & x\in\Omega, \, t > 0,\\
\nabla \cdot u &=& 0, \qquad & x\in\Omega, \, t > 0,\end{array} \right.$$
 which has been proposed as a model for the spatio-temporal evolution of populations of swimming aerobic bacteria. It is known that in bounded convex domains $${\Omega \subset \mathbb{R}^2}$$
 and under appropriate assumptions on the parameter functions χ, f and ϕ, for each $${\kappa\in\mathbb{R}}$$
 and all sufficiently smooth initial data this problem possesses a unique global-in-time classical solution. The present work asserts that this solution stabilizes to the spatially uniform equilibrium $${(\overline{n_0},0,0)}$$
 , where $${\overline{n_0}:=\frac{1}{|\Omega|} \int_\Omega n(x,0)\,{\rm d}x}$$
 , in the sense that as t→∞,

 $$n(\cdot,t) \to \overline{n_0}, \qquad c(\cdot,t) \to 0 \qquad \text{and}\qquad u(\cdot,t) \to 0$$
 hold with respect to the norm in $${L^\infty(\Omega)}$$
 .

Stabilization in a two-dimensional chemotaxis-Navier–Stokes system

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.

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

The chemotaxis-Navier-Stokes system    nt + u · ∇n = ∆n−∇ · (nχ(c)∇c), ct + u · ∇c = ∆c− nf(c), ut + (u · ∇)u = ∆u+∇P + n∇Φ, ∇ · u = 0, (⋆) (0.1) is considered under boundary conditions of homogeneous Neumann type for n and c, and Dirichlet type for u, in a bounded convex domain Ω ⊂ R with smooth boundary, where Φ ∈ W (Ω) and χ and f are sufficiently smooth given functions generalizing the prototypes χ ≡ const. and f(s) = s for s ≥ 0. It is known that for all suitably regular initial data n0, c0 and u0 satisfying 0 6≡ n0 ≥ 0, c0 ≥ 0 and ∇ · u0 = 0, 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 are ultimately satisfied. Moreover, it is shown that actually for any such eventual energy solution (n, c, u) there exists a waiting time T0 ∈ (0,∞) with the property that (n, c, u) is smooth in Ω× [T0,∞), and that n(x, t) → n0, c(x, t) → 0 and u(x, t) → 0 hold as t → ∞, uniformly with respect to x ∈ Ω. This resembles a classical result on the three-dimensional Navier-Stokes system, asserting eventual smoothness of arbitrary weak solutions thereof which additionally fulfill the associated natural energy inequality. In consequence, our results inter alia indicate that under the considered boundary conditions, the possibly destabilizing action of chemotactic cross-diffusion in (⋆) does not substantially affect the regularity properties of the fluid flow at least on large time scales.

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How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system?

Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion

Global existence and boundedness in a Keller–Segel–Stokes system involving a tensor-valued sensitivity with saturation: The 3D case

A Note on Global Existence for the Chemotaxis–Stokes Model with Nonlinear Diffusion

Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis

The chemotaxis-Navier-Stokes system with signal production, as given by 8 >< >: nt + u · rn = 1n r · nS(x, n, c) · rc ct + u · rc = 1c c + n ut + (u · r)u = 1u rP + nr r · u = 0, is considered in bounded planar convex domains with smooth boundary, where 2 W2,1() and S 2 C2(̄ ⇥ [0,1)2; R2⇥2). The main results assert that parallel to the case of the corresponding KellerSegel system obtained on neglecting u, any arbitrarily small algebraic saturation effect in the chemotactic sensitivity at large densities is sufficient to rule out any blow-up phenomenon. Indeed, under the assumption that there exist S0 0 and ↵ > 0 such that |S(x, n, c)|  S0 · (1+ n) ↵ for all x 2 ̄, n 0 and c 0, it is shown that for all suitably regular initial data an associated initial-boundary value problem possesses a globally defined bounded classical solution. The analysis is based on the consecutive identification of three energy-like functionals, the first among which involves a certain sublinear L p seminorm of n. Mathematics Subject Classification (2010): 92C17 (primary); 35Q30, 35K55, 35B65, 35Q92 (secondary).

Zhaoyin Xiang

Papers

Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion

Global existence and boundedness in a Keller–Segel–Stokes system involving a tensor-valued sensitivity with saturation: The 3D case

A Note on Global Existence for the Chemotaxis–Stokes Model with Nonlinear Diffusion

Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis

Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity