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.     

Classical solutions to Navier–Stokes equations for nonhomogeneous incompressible fluids with non-negative densities

We study the Navier–Stokes equations for nonhomogeneous incompressible fluids in a bounded domain Ω of R³. We first prove the existence and uniqueness of local classical solutions to the initial boundary value problem of linear Stokes equations and then we obtain the existence and uniqueness of local classical solutions to the Navier–Stokes equations with vacuum under the assumption that the data satisfies a natural compatibility condition.     

Steady flows of Jeffrey–Hamel type from the half-plane into an infinite channel. 1. Linearization on an antisymmetric solution

In this paper our objective is to provide physically reasonable solutions for the stationary Navier–Stokes equations in a two-dimensional domain with two outlets to infinity, a semi-strip Π₋ and a half-plane K. The same problem in an aperture domain, i.e. in a domain with two half-plane outlets to infinity, has been studied but only under symmetry restrictions on the data. Here, we assume that the main asymptotic term of the solution takes an antisymmetric form in K and apply the technique of weighted spaces with detached asymptotics, i.e. we use spaces where the functions have prescribed asymptotic forms in the outlets.After first showing that the corresponding Stokes problem admits a unique solution if and only if certain compatibility conditions are satisfied, we write the Navier–Stokes equations as a perturbation of the Stokes problem and the crucial compatibility condition as an algebraic equation by which the flux becomes determined. Assuming that the coefficient of the main (antisymmetric) asymptotic term of the solution in K does not vanish and that the data are sufficiently small, we use a contraction principle to solve the Navier–Stokes system coupled with the algebraic equation.Finally, we discuss the ill-posedness of the Navier–Stokes problem with prescribed flux.     

Structure of the set of stationary solutions of phase-lock equations in superconductivity

In previous article [M. Zhan, Phase-lock equations and its connections to Ginzburg–Landau equations of superconductivity, J. Nonlinear Anal. 42 (2000) 1063–1075], we introduced a system of equations (phase-lock equations) to model the superconductivity phenomena. We investigated its connection to Ginzburg–Landau equations and proved the existence and uniqueness of both weak and strong solutions. In this article, we study the steady-state problem associated with the phase-lock equations. We prove that the steady-state problem has multiple solutions and show that the solution set enjoys some structural properties as proved by Foias and Teman for the Navier–Stokes equations in [C. Foias, R. Teman, Structure of the set of stationary solutions of the Navier–Stokes equations, Commun. Pure Appl. Math. XXX (1977) 149–164].     

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.     

Controllability of systems of Stokes equations with one control force: existence of insensitizing controls

In this paper we establish some exact controllability results for systems of two parabolic equations of the Stokes kind. In a first part, we prove the existence of insensitizing controls for the L² norm of the solutions and the curl of solutions of linear Stokes equations. Then, in the limit case where one can expect null controllability to hold for a system of two Stokes equations (namely, when the coupling terms concern first and second order derivatives, respectively), we prove this result for some general couplings.     

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.     

Insensitizing controls with one vanishing component for the Navier–Stokes system

In this paper we prove the existence of insensitizing controls, having one vanishing component, for the local L²$L^2$-norm of the solutions of the Navier–Stokes system. This problem can be recast as a null controllability problem for a nonlinear cascade system. We first prove a controllability result, with controls having one vanishing component, for a linear problem. Then, by means of an inverse mapping theorem, we deduce the controllability for the cascade system.     

Some Estimates near the Boundary for Solutions to the Nonstationary Linearized Navier–Stokes Equations

In the present paper, new local estimates near the boundary are established for solutions to the nonstationary linearized Navier–Stokes equations are established. Bibliography: 8 titles.     

Existence and Regularity of the Pressure for the Stochastic Navier–Stokes Equations

We prove, on one hand, that for a convenient body force with values in the distribution space (H ^-1(D)) ^d , where D is the geometric domain of the fluid, there exist a velocity u and a pressure p solution of the stochastic Navier–Stokes equation in dimension 2, 3 or 4. On the other hand, we prove that, for a body force with values in the dual space V of the divergence free subspace V of (H ¹ ₀(D)) ^d , in general it is not possible to solve the stochastic Navier–Stokes equations. More precisely, although such body forces have been considered, there is no topological space in which Navier–Stokes equations could be meaningful for them.     

Quasistationary Approximation in the Rotating Ring Problem

We consider the planar rotation-symmetric motion by inertia of a viscous incompressible fluid in a ring with free boundary. We reduce the corresponding initial-boundary value problem for the Navier–Stokes equations to some problem for a coupled system of one parabolic equation and two ordinary differential equations. We suppose that the coefficient of the derivatives of the sought functions with respect to time (the quasistationary parameter) is small; so the system is singularly perturbed. In this article we construct an asymptotic expansion for a solution to the rotating ring problem in a small quasistationary parameter and obtain a smallness estimate for the difference between the exact and approximate solutions.     

Review and complements on mixed-hybrid finite element methods for fluid flows

Mixed and hybrid finite element methods for the resolution of a wide range of linear and nonlinear boundary value problems (linear elasticity, Stokes problem, Navier–Stokes equations, Boussinesq equations, etc.) have known a great development in the last few years. These methods allow simultaneous computation of the original variable and its gradient, both of them being equally accurate. Moreover, they have local conservation properties (conservation of the mass and the momentum) as in the finite volume methods.The purpose of this paper is to give a review on some mixed finite elements developed recently for the resolution of Stokes and Navier–Stokes equations, and the linear elasticity problem. Further developments for a quasi-Newtonian flow obeying the power law are presented.     

Approximate controllability for linearized compressible barotropic Navier–Stokes system in one and two dimensions

In this paper we consider compressible barotropic Navier–Stokes equations in one and two dimensions linearized around a constant steady state with Dirichlet boundary conditions. We explore the controllability of this linearized system using a control only for the velocity equation. We prove that the system with homogeneous Dirichlet boundary conditions, is approximately controllable by a localized interior control when time is sufficiently large.     

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=∞).     

Remarks on exact controllability for Stokes and Navier–Stokes systems

This Note deals with the controllability of Stokes and Navier–Stokes systems with distributed controls with support in possibly small subdomains. We first present a new global Carleman inequality for the solutions to Stokes-like systems that leads to the null controllability at any time T>0. Then, we present a local result concerning exact controllability to trajectories of the Navier–Stokes system. To cite this article: E. Fernández-Cara et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

A Phragmén–Lindelöf alternative result for the Navier–Stokes equations for steady compressible viscous flow

In this paper, the authors consider the Navier–Stokes equations for steady compressible viscous flow in three-dimensional cylindrical domain. A differential inequality for appropriate energy associated with the solutions of the Navier–Stokes isentropic flow in semi-infinite pipe is derived, from which the authors show a Phragmén–Lindelöf alternative result, i.e. the solutions for steady compressible viscous N–S flow problem either grow or decay exponentially as the distance from the entry section tends to infinity. In the decay case, the authors indicate how to bound explicitly the total energy in terms of data.     

A two-grid method based on Newton iteration for the Navier–Stokes equations

In this paper, we consider a two-grid method for resolving the nonlinearity in finite element approximations of the equilibrium Navier–Stokes equations. We prove the convergence rate of the approximation obtained by this method. The two-grid method involves solving one small, nonlinear coarse mesh system and two linear problems on the fine mesh which have the same stiffness matrix with only different right-hand side. The algorithm we study produces an approximate solution with the optimal asymptotic in h and accuracy for any Reynolds number. Numerical example is given to show the convergence of the method.     

Numerical comparison of nonlinear subgridscale models via adaptive mesh refinement

In this paper, we provide a method of evaluating the efficacy of nonlinear subgridscale models for use in the large eddy simulation of incompressible viscous flow problems. We compare subgridscale “artificial” viscosity models using a posteriori error estimation and adaptive mesh refinement. Specifically, we compare α-Laplacian based subgridscale models and discuss the benefits and limitations of different values of α for some standard benchmark problems for the Navier–Stokes equations.     

The Navier–Stokes limit of the Boltzmann equation for bounded collision kernels

The present work establishes a Navier–Stokes limit for the Boltzmann equation considered over the infinite spatial domain R ³. Appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations whose limit points (in the w-L ¹ topology) are governed by Leray solutions of the limiting Navier–Stokes equations. This completes the arguments in Bardos-Golse-Levermore [Commun. Pure Appl. Math. 46(5), 667–753 (1993)] for the steady case, and in Lions-Masmoudi [Arch. Ration. Mech. Anal. 158(3), 173–193 (2001)] for the time-dependent case.Mathematics Subject Classification (2000) 35Q35, 35Q30, 82C40