Time-dependent Stokes and Navier–Stokes problems with boundary conditions involving pressure, existence and regularity

This paper is concerned with the time-dependent Stokes and Navier–Stokes problems with nonstandard boundary conditions: the pressure is given on some part of the boundary. The stationary case was first studied by Bégue, Conca, Murat and Pironneau and, next, their study were completed by Bernard, mainly about regularity. In this paper, the Stokes problem is studied by a method analogous to that of Temam for the standard problem, combined with regularity results of Bernard for the nonstandard stationary case. We obtain existence, uniqueness and regularity H². In addition, in two dimensions, a regularity W^2,r, r2, is proved. Next, for the nonstandard Navier–Stokes problem, we present some existence, uniqueness and regularity H² results. The proof of existence is based on a fixed point method.     

Coupling method for the exterior stationary Navier–Stokes equations

We represent a new numerical method to solve the stationary Navier–Stokes equations in an unbounded domain. This technique consists in coupling the boundary integral and finite element methods. Moreover, we derive the variational formulation and well-posedness of the coupling method and provide the convergence result for the approximate solution. © 1993 John Wiley & Sons, Inc.     

Two-level Galerkin–Lagrange multipliers method for the stationary Navier–Stokes equations

In this paper, we combine the Galerkin–Lagrange multiplier (GLM) method with the two-level method to solve the stationary Navier–Stokes equations in order to avoid the time-consuming process and the construction of zero-divergence elements. Different quadrilateral partitions are used for approximating the velocity and the pressure. Then some error estimates are obtained and some numerical results of the GLM method and the two-level GLM method are given. The results show that the two-level method based on the GLM method is more efficient than the GLM method under the convergence rate of same order.     

A preconditioner for generalized saddle point problems: Application to 3D stationary Navier‐Stokes equations

In this article we consider the stationary Navier‐Stokes system discretized by finite element methods which do not satisfy the inf‐sup condition. These discretizations typically take the form of a variational problem with stabilization terms. Such a problem may be transformed by iteration methods into a sequence of linear, Oseen‐type variational problems. On the algebraic level, these problems belong to a certain class of linear systems with nonsymmetric system matrices (“generalized saddle point problems”). We show that if the underlying finite element spaces satisfy a generalized inf‐sup condition, these problems have a unique solution. Moreover, we introduce a block triangular preconditioner and we show how the eigenvalue bounds of the preconditioned system matrix depend on the coercivity constant and continuity bounds of the bilinear forms arising in the variational problem. Finally we prove that the stabilized P1‐P1 finite element method proposed by Rebollo is covered by our theory and we show that the condition number of the preconditioned system matrix is independent of the mesh size. Numerical tests with 3D stationary Navier‐Stokes flows confirm our results. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006     

The stationary three‐dimensional Navier–Stokes equations with a non‐zero constant velocity at infinity

This paper is devoted to some mathematical questions related to the three‐dimensional stationary Navier–Stokes equations. Our approach is based on a combination of properties of Oseen problems in ?³. Copyright © 2008 John Wiley & Sons, Ltd.     

Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions

In this paper, we study the existence and regularity of solutions to the Stokes and Oseen equations with nonhomogeneous Dirichlet boundary conditions with low regularity. We consider boundary conditions for which the normal component is not equal to zero. We rewrite the Stokes and the Oseen equations in the form of a system of two equations. The first one is an evolution equation satisfied by Pu, the projection of the solution on the Stokes space – the space of divergence free vector fields with a normal trace equal to zero – and the second one is a quasi-stationary elliptic equation satisfied by (I−P)u, the projection of the solution on the orthogonal complement of the Stokes space. We establish optimal regularity results for Pu and (I−P)u. We also study the existence of weak solutions to the three-dimensional instationary Navier–Stokes equations for more regular data, but without any smallness assumption on the initial and boundary conditions.     

Stability Analysis of Navier–Stokes Flows in Annuli

We study the linearized stability of planer flows of incompressible, viscous fluid in two-dimensional annular domains. A certain family of steady, explicit solutions which have spiral streamlines are considered. The Navier–Stokes equations are linearized at these solutions and we show analytically or numerically that these solutions are stable to perturbation of steady states, whatever the Reynolds number or the aspect ratio of the annuli may be. Hopf bifurcations from them are also examined numerically. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Local exact controllability of the Navier–Stokes system

In this paper we deal with the local exact controllability of the Navier–Stokes system with distributed controls supported in small sets. In a first step, we present a new Carleman inequality for the linearized Navier–Stokes system, which leads to null controllability at any time T>0. Then, we deduce a local result concerning the exact controllability to the trajectories of the Navier–Stokes system.     

A note on integration of the Navier–Stokes equations

In this study the 2D Navier–Stokes equations are used to obtain a new self-similar equation. The latter equation, subject to appropriate boundary conditions and volume discharge, describes the pressure distribution and velocity field of a plane free jet.     

A stabilized finite element method based on local polynomial pressure projection for the stationary Navier–Stokes equations

This article considers a stabilized finite element approximation for the branch of nonsingular solutions of the stationary Navier–Stokes equations based on local polynomial pressure projection by using the lowest equal-order elements. The proposed stabilized method has a number of attractive computational properties. Firstly, it is free from stabilization parameters. Secondly, it only requires the simple and efficient calculation of Gauss integral residual terms. Thirdly, it can be implemented at the element level. The optimal error estimate is obtained by the standard finite element technique. Finally, comparison with other methods, through a series of numerical experiments, shows that this method has better stability and accuracy.     

Stabilized spectral element approximation for the Navier–Stokes equations

The conforming spectral element methods are applied to solve the linearized Navier–Stokes equations by the help of stabilization techniques like those applied for finite elements. The stability and convergence analysis is carried out and essential numerical results are presented demonstrating the high accuracy of the method as well as its robustness. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 115–141, 1998     

Strong solutions for the nonhomogeneous Navier–Stokes equations in unbounded domains

We show the existence of strong solutions for the nonhomogeneous Navier–Stokes equations in three‐dimensional domains with boundary uniformly of class C³. Under suitable assumptions, uniqueness is also proved. Copyright © 2009 John Wiley & Sons, Ltd.     

Feedback boundary stabilization of the three-dimensional incompressible Navier–Stokes equations

We study the local stabilization of the three-dimensional Navier–Stokes equations around an unstable stationary solution w, by means of a feedback boundary control. We first determine a feedback law for the linearized system around w. Next, we show that this feedback provides a local stabilization of the Navier–Stokes equations. To deal with the nonlinear term, the solutions to the closed loop system must be in H^{3/2+ε,3/4+ε/2}(Q), with 0<ε. In [V. Barbu, I. Lasiecka, R. Triggiani, Boundary stabilization of Navier–Stokes equations, Mem. Amer. Math. Soc. 852 (2006); V. Barbu, I. Lasiecka, R. Triggiani, Abstract settings for tangential boundary stabilization of Navier–Stokes equations by high- and low-gain feedback controllers, Nonlinear Anal. 64 (2006) 2704–2746], such a regularity is achieved with a feedback obtained by minimizing a functional involving a norm of the state variable strong enough. In that case, the feedback controller cannot be determined by a well posed Riccati equation. Here, we choose a functional involving a very weak norm of the state variable. The compatibility condition between the initial state and the feedback controller at t=0, is achieved by choosing a time varying control operator in a neighbourhood of t=0.     

Abstract Besov space approach to the non-stationary Navier–Stokes equations

The local existence, uniqueness and regularity of solutions of the initial-value problem for non-stationary Navier–Stokes equations are studied via abstract Besov spaces. The assumptions about Ω the initial data and the external force are improved on by rather simple proofs.     

Leading edge separation: a comparison between interaction theory and Navier–Stokes computations

We consider the two‐dimensional high/moderate Reynolds‐number flow around a slender airfoil at small angles of attack. Special emphasis is placed on leading edge separation and the associated breakdown of the steady solution of the governing equations. A comparison between the predictions of the viscous/inviscid interaction theory of marginal separation, see e.g. [3], and the results of Navier–Stokes computations with DLR's ‘TAU’–code [2] shows excellent quantitative agreement concerning the location, size of the separation bubble and even the limiting maximum value of the angle of attack, up to which solutions by means of the particular approaches can be found.     

Non‐local approach in mathematical problems of fluid–structure interaction

Three‐dimensional mathematical problems of interaction between elastic and scalar oscillation fields are investigated. An elastic field is to be defined in a bounded inhomogeneous anisotropic body occupying the domain Ω¯₁⊂ℝ³ while a physical (acoustic) scalar field is to be defined in the exterior domain Ω¯₂=ℝ³\Ω₁ which is filled up also by an anisotropic (fluid) medium. These two fields satisfy the governing equations of steady‐state oscillations in the corresponding domains together with special kinematic and dynamic transmission conditions on the interface ∂Ω₁. The problems are studied by the so‐called non‐local approach, which is the coupling of the boundary integral equation method (in the unbounded domain) and the functional‐variational method (in the bounded domain). The uniqueness and existence theorems are proved and the regularity of solutions are established with the help of the corresponding Steklov–Poincaré type operators and on the basis of the Gårding inequality and the Lax–Milgram theorem. In particular, it is shown that the physical fluid–solid acoustic interaction problem is solvable for arbitrary values of the frequency parameter. Copyright © 1999 John Wiley & Sons, Ltd.     

Strong Solutions of the Navier–Stokes System in Lipschitz Bounded Domains

Let Ω be a bounded domain in ?³ with a connected Lipschitz boundary ?Ω. Consider the Navier-Stokes system.     

On missing boundary conditions with unsteady incompressible Navier–Stokes flows

We investigate an unsteady viscous flow problem where ‘good’ boundary conditions are available on part of the boundary only. This problem appears when the flow phenomena one is interested in are concentrated on part of the flow region and, for reasons of computational economy, are numerically computed in this subregion only. Assuming that outside of the subregion the flow is not subjected to any acceleration forces, we develop an (abstract) combined finite-element/boundary element scheme to compute the flow approximately. This scheme leads to a proof of the existence of a weak solution of the corresponding Navier–Stokes problem as well.     

Asymptotic behavior for the Navier–Stokes equations in 2D exterior domains

We show that the L^p spatial–temporal decay rates of solutions of incompressible flow in an 2D exterior domain. When a domain has a boundary, pressure term makes an obstacle since we do not have enough information on the pressure term near the boundary. To overcome the difficulty, we adopt the ideas in He, Xin [C. He, Z. Xin, Weighted estimates for nonstationary Navier–Stokes equations in exterior domain, Methods Appl. Anal. 7 (3) (2000) 443–458], and our previous results [H.-O. Bae, B.J. Jin, Asymptotic behavior of Stokes solutions in 2D exterior domains, J. Math. Fluid Mech., in press; H.-O. Bae, B.J. Jin, Temporal and spatial decay rates of Navier–Stokes solutions in exterior domains, submitted for publication]. For the spatial decay rate estimate, we first extend temporal decay rate result of the Navier–Stokes solutions for general L^p space when the initial velocity is in , 1<rq<∞ (1<r<q=∞).