Quasi-neutral and zero-viscosity limits of Navier–Stokes–Poisson equations in the half-space

The present paper is concerned with the quasi-neutral and zero-viscosity limits of Navier–Stokes–Poisson equations in the half-space. We consider the Navier-slip boundary condition for velocity and Dirichlet boundary condition for electric potential. By means of asymptotic analysis with multiple scales, we construct an approximate solution of the Navier–Stokes–Poisson equations involving two different kinds of boundary layer, and establish the linear stability of the boundary layer approximations by conormal energy estimate.     

Convergence of the quantum Navier–Stokes–Poisson equations to the incompressible Euler equations for general initial data

In this paper, we are concerned with the rigorous proof of the convergence of the quantum Navier–Stokes-Poisson system to the incompressible Euler equations via the combined quasi-neutral, vanishing damping coefficient and inviscid limits in the three-dimensional torus for general initial data. Furthermore, the convergence rates are obtained.     

Stability of the nonconstant stationary solution to the Poisson–Nernst–Planck–Navier–Stokes equations

We study the three-dimensional Cauchy problem of the Poisson–Nernst–Planck–Navier–Stokes equations. We first show that the corresponding stationary system has a unique semi-trivial solution under a general doping profile. Under initial small perturbations around such the semi-trivial stationary solution and small doping profile, we obtain the unique global-in-time solution to the non-stationary system. Moreover, we prove the asymptotic convergence of the solution toward the semi-trivial stationary solution as time tends to infinity.     

Approximations of stochastic Navier–Stokes equations

In this paper we show that solutions of two-dimensional stochastic Navier–Stokes equations driven by Brownian motion can be approximated by stochastic Navier–Stokes equations forced by pure jump noise/random kicks.     

On some properties of the Navier–Stokes system of equations

We discuss the initial and boundary value problems for the system of dimensionless Navier–Stokes equations describing the dynamics of a viscous incompressible fluid using the method of characteristics and the geometric method developed by the authors. Some properties of the formulation of these problems are considered. We study the effect of the Reynolds number on the flow of a viscous fluid near the surface of a body.     

Asymptotic compactness of global trajectories generated by the Navier–Stokes–Poisson equations of a compressible fluid

In this paper, we consider the global behavior of weak solutions of Navier–Stokes–Poisson equations in time in a bounded domain–arbitrary forces. After proving the existence of bounded absorbing sets, we also obtain the conclusion on asymptotic compactness of global trajectories generated by the Navier–Stokes–Poisson equations of a compressible fluid. Supported by NSF(No:10531020) of China and the Program of 985 Innovation Engineering on Information in Xiamen University (2004–2007).     

Navier–Stokes equations,the algebraic aspect

Theoretical and Mathematical Physics - We present an analysis of the Navier–Stokes equations in the framework of an algebraic approach to systems of partial differential equations (the formal...     

Self-similar solutions of stationary Navier–Stokes equations

In this paper, we mainly study the existence of self-similar solutions of stationary Navier–Stokes equations for dimension $n=3,4$. For $n=3$, if the external force is axisymmetric, scaling invariant, $C^{1,\alpha }$ continuous away from the origin and small enough on the sphere $S^2$, we shall prove that there exists a family of axisymmetric self-similar solutions which can be arbitrarily large in the class $C_{loc}^{3,\alpha }({\mathbb{R}}^3\backslash 0)$. Moreover, for axisymmetric external forces without swirl, corresponding to this family, the momentum flux of the flow along the symmetry axis can take any real number. However, there are no regular ($U\in C_{loc}^{3,\alpha }({\mathbb{R}}^3\backslash 0)$) axisymmetric self-similar solutions provided that the external force is a large multiple of some scaling invariant axisymmetric F which cannot be driven by a potential. In the case of dimension 4, there always exists at least one self-similar solution to the stationary Navier–Stokes equations with any scaling invariant external force in $L^{4/3,\infty }({\mathbb{R}}^4)$.     

Analysis of the Yosida method for the incompressible Navier–Stokes equations

The Yosida method was introduced in (Quarteroni et al., to appear) for the numerical approximation of the incompressible unsteady Navier–Stokes equations. From the algebraic viewpoint, it can be regarded as an inexact factorization of the matrix arising from the space and time discretization of the problem. However, its differential interpretation resides on an elliptic stabilization of the continuity equation through the Yosida regularization of the Laplacian (see (Brezis, 1983, Ciarlet and Lions, 1991)). The motivation of this method as well as an extensive numerical validation were given in (Quarteroni et al., to appear).In this paper we carry out the analysis of this scheme. In particular, we consider a first-order time advancing unsplit method. In the case of the Stokes problem, we prove unconditional stability and moreover that the splitting error introduced by the Yosida scheme does not affect the overall accuracy of the solution, which remains linear with respect to the time step. Some numerical experiments, for both the Stokes and Navier–Stokes equations, are presented in order to substantiate our theoretical results.     

On very weak solutions of the stationary Navier–Stokes equations

We study the stationary Navier–Stokes equations in a bounded domain Ω of R ³ with smooth connected boundary. The notion of very weak solutions has been introduced by Marušić-Paloka (Appl. Math. Optim. 41:365–375, 2000), Galdi et al. (Math. Ann. 331:41–74, 2005) and Kim (Arch. Ration. Mech. Anal. 193:117–152, 2009) to obtain solvability results for the Navier–Stokes equations with very irregular data. In this article, we prove a complete solvability result which unifies those in Marušić-Paloka (Appl. Math. Optim. 41:365–375, 2000), Galdi et al. (Math. Ann. 331:41–74, 2005) and Kim (Arch. Ration. Mech. Anal. 193:117–152, 2009) by adapting the arguments in Choe and Kim (Preprint) and Kim and Kozono (Preprint).     

On Landau’s solutions of the Navier–Stokes equations

A special class of solutions of the n-dimensional steady-state Navier–Stokes equations is considered. Bibliography: 23 titles.     

On symmetric Leray solutions of the stationary Navier–Stokes equations

A classical result of Amick (Acta Math 161:71–130, 1988) on the nontriviality of the symmetric Leray solutions of the steady-state Navier–Stokes equations in the plane is extended to Lipschitz domains. This results is compared with the famous Stokes paradox of linearized hydrodynamics and applied to a mixed problem of some interest in the applications.     

A reliable approach to the solution of Navier–Stokes equations

In this paper we will demonstrate an affective approach of solving Navier–Stokes equations by using a very reliable transformation method known as the Cole–Hopf transformation, which reduces the problem from nonlinear into a linear differential equation which, in turn, can be solved effectively.