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.     

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.     

The convergence of a new parallel algorithm for the Navier–Stokes equations

This paper deals with the convergence and stability of a new parallel algorithm and the error estimates for a particular case of the new parallel algorithm, which is used to solve the incompressible nonstationary Navier–Stokes equations. The theoretical results show that the scheme is (at least) conditionally stable and convergent.     

The approximation of a Crank–Nicolson scheme for the stochastic Navier–Stokes equations

In this paper we prove the convergence of stochastic Navier–Stokes equations driven by white noise. A linearized version of the implicit Crank–Nicolson scheme is considered for the approximation of the solutions to the N–S equations. The noise is defined as the distributional derivative of a Wiener process and approximated by using the generalized _L2$L_2$-projection operator. Optimal strong convergence error estimates in the _L2$L_2$ norm are obtained.     

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.     

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 $L^{4/3,\infty }({\mathbb{R}}^4)$.     

Necessary conditions of a potential blow-up for Navier–Stokes equations

Assuming that T is a potential blow-up time, it is shown that the H^frac12 {H^{frac{1}{2}}} -norm of the velocity field goes to ∞ as the time t approaches T. Bibliography: 9 titles.     

Boundary layer problem: Navier–Stokes equations and Euler equations

This work is concerned with the boundary layer turbulence, which is an outstanding problem in fluid mechanics. We consider an incompressible viscous fluid in 2D domains with permeable walls. The permeability is described by the Yudovich condition. The goal of the article is to study the fluid behavior at vanishing viscosity (large Reynold’s numbers). We show that the vanishing viscous limit is a solution of the Euler equations with the Yudovich condition on the inflow region of the boundary.     

Existence of stationary solutions of the Navier–Stokes equations in the presence of a wall

We consider the problem of a body moving within an incompressible fluid at constant speed parallel to a wall, in an otherwise unbounded domain. This situation is modeled by the incompressible Navier–Stokes equations in an exterior domain in a half space, with appropriate boundary conditions on the wall, the body, and at infinity. Here, we prove existence of stationary solutions for this problem for the simplified situation where the body is replaced by a source term of compact support.     

The analysis of a finite element method for the Navier–Stokes equations with compressibility

Summary. A finite element formulation is developed for the two dimensional nonlinear time dependent compressible Navier–Stokes equations on a bounded domain. The existence and uniqueness of the solution to the numerical formulation is proved. An error estimate for the numerical solution is obtained. Received September 9, 1997 / Revised version received August 12, 1999 / Published online July 12, 2000     

Navier–Stokes equations and forward–backward SDEs on the group of diffeomorphisms of a torus

We establish a connection between the strong solution to the spatially periodic Navier–Stokes equations and a solution to a system of forward–backward stochastic differential equations (FBSDEs) on the group of volume-preserving diffeomorphisms of a flat torus. We construct representations of the strong solution to the Navier–Stokes equations in terms of diffusion processes.     

Derivation of Prandtl boundary layer equations for the incompressible Navier–Stokes equations in a curved domain

The proposal of this note is to derive the equations of boundary layers in the small viscosity limit for the two-dimensional incompressible Navier–Stokes equations defined in a curved bounded domain with the non-slip boundary condition. By using curvilinear coordinate system in a neighborhood of boundary, and the multi-scale analysis we deduce that the leading profiles of boundary layers of the incompressible flows in a bounded domain still satisfy the classical Prandtl equations when the viscosity goes to zero, which are the same as for the flows defined in the half space.     

Nonexistence of asymptotically self-similar singularities in the Euler and the Navier–Stokes equations

In this paper we rule out the possibility of asymptotically self-similar singularities for both of the 3D Euler and the 3D Navier–Stokes equations. The notion means that the local in time classical solutions of the equations develop self-similar profiles as t goes to the possible time of singularity T. For the Euler equations we consider the case where the vorticity converges to the corresponding self-similar voriticity profile in the sense of the critical Besov space norm, . For the Navier–Stokes equations the convergence of the velocity to the self-similar singularity is in L ^q (B(z,r)) for some , where the ball of radius r is shrinking toward a possible singularity point z at the order of as t approaches to T. In the convergence case with we present a simple alternative proof of the similar result in Hou and Li in arXiv-preprint, math.AP/0603126. This work was supported partially by KRF Grant(MOEHRD, Basic Research Promotion Fund) and the KOSEF Grant no. R01-2005-000-10077-0.