High-order accurate iterative solution of the Navier–Stokes equations for incompressible flows

An iterative solution scheme for the incompressible Navier–Stokes equations is presented. It is split into inner and outer iteration cycles, such that the momentum and continuity equations are satisfied within prescribed accuracy. The spatial discretization is based on high-order finite differences which makes it well suited for massively parallel computers. This is demonstrated in a scaling test. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Exact controllability in projections for three-dimensional Navier–Stokes equations

The paper is devoted to studying controllability properties for 3D Navier–Stokes equations in a bounded domain. We establish a sufficient condition under which the problem in question is exactly controllable in any finite-dimensional projection. Our sufficient condition is verified for any torus in R³${\mathbb{R}}^3$. The proofs are based on a development of a general approach introduced by Agrachev and Sarychev in the 2D case. As a simple consequence of the result on controllability, we show that the Cauchy problem for the 3D Navier–Stokes system has a unique strong solution for any initial function and a large class of external forces.     

Local regularity for suitable weak solutions of Navier–Stokes equations near the boundary

A class of sufficient conditions for the local boundary regularity of suitable weak solutions of nonstationary three-dimensional Navier–Stokes equations is discussed. The corresponding results are stated in terms of functionals invariant with respect to the scaling of Navier–Stokes equations. Bibliography: 26 titles.     

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.     

On a problem for the Navier–Stokes equations with the infinite Dirichlet integral

We study existence of global in time solutions to the Navier–Stokes equations in a two dimensional domain with an unbounded boundary. The problem is considered with slip boundary conditions involving nonzero friction. The main result shows a new L-bound on the vorticity. A key element of the proof is the maximum principle for a reformulation of the problem. Under some restrictions on the curvature of the boundary and the friction the result for large data (including flux) with the infinite Dirichlet integral is obtained.Received: October 31, 2002; revised: September 17, 2003     

On a problem for the Navier–Stokes equations with the infinite Dirichlet integral

We study existence of global in time solutions to the Navier–Stokes equations in a two dimensional domain with an unbounded boundary. The problem is considered with slip boundary conditions involving nonzero friction. The main result shows a new L-bound on the vorticity. A key element of the proof is the maximum principle for a reformulation of the problem. Under some restrictions on the curvature of the boundary and the friction the result for large data (including flux) with the infinite Dirichlet integral is obtained.     

Existence of solutions for a class of Navier–Stokes equations with infinite delay

In this paper, a class of Navier–Stokes equations with infinite delay is considered. It includes delays in the convective and the forcing terms. We discuss the existence of mild and classical solutions for the problem. We establish the results for an abstract delay problem by using the fact that the Stokes operator is the infinitesimal generator of an analytic semigroup of bounded linear operators. Finally, we apply these abstract results to our particular situation.     

Existence of global stochastic flow and attractors for Navier–Stokes equations

For 2-D stochastic Navier-Stokes equations on the torus with multiplicative noise we construct a perfect cocycle and show the existence of global random compact attractors. The equations considered do not admit a pathwise method of solution. Received: 9 June 1998 / Revised version: 17 December 1998     

Local exact controllability for the two- and three-dimensional compressible Navier–Stokes equations

The goal of this article is to present a local exact controllability result for the two- and three-dimensional compressible Navier–Stokes equations on a constant target trajectory when the controls act on the whole boundary. Our study is then based on the observability of the adjoint system of some linearized version of the system, which is analyzed using a subsystem for which the coupling terms are somewhat weaker. In this step, we strongly use Carleman estimates in negative Sobolev spaces.     

On local regularity for suitable weak solutions of the Navier–Stokes equations near the boundary

A class of sufficient conditions for local boundary regularity of suitable weak solutions of nonstationary three-dimensional Navier–Stokes equations is discussed. The corresponding results are stated in terms of functionals, which are invariant with respect to the scaling of the Navier–Stokes equations. Bibliography: 27 titles.     

Stability behaviors of Leray weak solutions to the three-dimensional Navier–Stokes equations

This paper is devoted to the investigation of stability behaviors of Leray weak solutions to the three-dimensional Navier–Stokes equations. For a Leray weak solution of the Navier–Stokes equations in a critical Besov space, it is shown that the Leray weak solution is uniformly stable with respect to a small perturbation of initial velocity and external forcing. If the perturbation is not small, the perturbed weak solution converges asymptotically to the original weak solution as the time tends to the infinity. Additionally, an energy equality and weak–strong uniqueness for the three-dimensional Navier–Stokes equations are derived. The findings are mainly based on the estimations of the nonlinear term of the Navier–Stokes equations in a Besov space framework, the use of special test functions and the energy estimate method.     

Long-time behavior of solution for the compressible Navier–Stokes–Korteweg equations in R3

In this paper, we study the long-time behavior of solution for the compressible Navier–Stokes–Korteweg equations in three-dimensional whole space. More precisely, we focus on establishing the optimal time decay rates for the higher-order spatial derivatives of density and velocity, which will improve the work of Wang and Tan (2011).     

Convergence analysis of an implicit fractional-step method for the incompressible Navier–Stokes equations

In this paper, an implicit fractional-step method for numerical solutions of the incompressible Navier–Stokes equations is studied. The time advancement is decomposed into a sequence of two steps, and the first step can be seen as a linear elliptic problem; on the other hand, the second step has the structure of the Stokes problem. The two problems satisfy the full homogeneous Dirichlet boundary conditions on the velocity. At the same time, we introduce a diffusion term −θΔu in all steps of the schemes. It allows to calculate by the large time step and enhance numerical stability by choosing the proper parameter values of θ. The convergence analysis and error estimates for the intermediate velocities, the end-of step velocities and the pressure solution are derived. Finally, numerical experiments show that the feasibility and effectiveness of this method.     

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.