Bilinear estimates in BMO and the Navier-Stokes equations

We prove that the BMO norm of the velocity and the vorticity controls the blow-up phenomena of smooth solutions to the Navier-Stokes equations. Our result is applied to the criterion on uniqueness and regularity of weak solutions in the marginal class. Received February 15, 1999; in final form October 11, 1999 / Published online July 3, 2000     

Rapid time-decay and net force to the obstacles by the Stokes flow in exterior domains

Consider the nonstationary Stokes equations in exterior domains with the compact boundary . We show first that the solution decays like for all as . This decay rate is optimal in the sense that for some as occurs if and only if the net force exerted by the fluid on is zero. Received: 15 June 2000 / Published online: 18 June 2001     

Convergence and stability of finite element nonlinear Galerkin method for the Navier-Stokes equations

In this article, we present a new fully discrete finite element nonlinear Galerkin method, which are well suited to the long time integration of the Navier-Stokes equations. Spatial discretization is based on two-grid finite element technique; time discretization is based on Euler explicit scheme with variable time step size. Moreover, we analyse the boundedness, convergence and stability condition of the finite element nonlinear Galerkin method. Our discussion shows that the time step constraints of the method depend only on the coarse grid parameter and the time step constraints of the finite element Galerkin method depend on the fine grid parameter under the same convergence accuracy. Received February 2, 1994 / Revised version received December 6, 1996     

On the zero-velocity-limit solutions to the Navier–Stokes equations of compressible flow

We consider the zero-velocity stationary problem of the Navier–Stokes equations of compressible isentropic flow describing the distribution of the density ϱ of a fluid in a spatial domain Ω⊂ℝ ^N driven by a time-independent potential external force b=∇F. A sharp condition in terms of F is given for the problem to possess a unique nonnegative solution ϱ having a prescribed mass m > 0. Received: 20 October 1997     

Error indicators for the Navier-Stokes equations in stream function and vorticity formulation

This paper deals with a posteriori estimates for the finite element solution of the Stokes problem in stream function and vorticity formulation. For two different discretizations, we propose error indicators and we prove estimates in order to compare them with the local error. In a second step, these results are extended to the Navier-Stokes equations. Received March 25, 1996 / Revised version received April 7, 1997     

Weighted regularization of Maxwell equations in polyhedral domains

Summary. We present a new method of regularizing time harmonic Maxwell equations by a {\bf grad}-div term adapted to the geometry of the domain. This method applies to polygonal domains in two dimensions as well as to polyhedral domains in three dimensions. In the presence of reentrant corners or edges, the usual regularization is known to produce wrong solutions due the non-density of smooth fields in the variational space. We get rid of this undesirable effect by the introduction of special weights inside the divergence integral. Standard finite elements can then be used for the approximation of the solution. This method proves to be numerically efficient. Received April 27, 2001 / Revised version received September 13, 2001 / Published online March 8, 2002     

Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations

Summary. We examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady-state Navier-Stokes equations. With a combination of analytic and empirical results, we study the effects of fundamental parameters on convergence. We demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. The structure of these distributions is independent of the discretization mesh size, but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller. These characteristics are directly correlated with the convergence properties of iterative solvers. Received August 5, 2000 / Published online June 20, 2001     

Analyticity and smoothing effect for the Korteweg de Vries equation with a single point singularity

We show that a solution of the Cauchy problem for the KdV equation, has a drastic smoothing effect up to real analyticity if the initial data only have a single point singularity at x = 0. It is shown that for () data satisfying the condition the solution is analytic in both space and time variable. The above condition allows us to take as initial data the Dirac measure or the Cauchy principal value of 1/x. The argument is based on the recent progress on the well-posedness result by Bourgain [2] and Kenig-Ponce-Vega [20] and a systematic use of the dilation generator . Received 22 March 1999     

Convergence analysis and error estimates for a parallel algorithm for solving the Navier-Stokes equations

Summary. This paper is concerned with the analysis of the convergence and the derivation of error estimates for a parallel algorithm which is used to solve the incompressible Navier-Stokes equations. As usual, the main idea is to split the main differential operator; this allows to consider independently the two main difficulties, namely nonlinearity and incompressibility. The results justify the observed accuracy of related numerical results. Received April 20, 2001 / Revised version received May 21, 2001 / Published online March 8, 2002 RID="*" ID="*" Partially supported by D.G.E.S. (Spain), Proyecto PB98–1134 RID="**" ID="**" Partially supported by D.G.E.S. (Spain), Proyecto PB96–0986 RID="**" ID="**" Partially supported by D.G.E.S. (Spain), Proyecto PB96–0986 RID="*" ID="*" Partially supported by D.G.E.S. (Spain), Proyecto PB98–1134 RID="**" ID="**" Partially supported by D.G.E.S. (Spain), Proyecto PB96–0986 RID="**" ID="**" Partially supported by D.G.E.S. (Spain) Proyecto PB96–0986     

Nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the Navier-Stokes equations

Summary. A nonlinear Galerkin method using mixed finite elements is presented for the two-dimensional incompressible Navier-Stokes equations. The scheme is based on two finite element spaces and for the approximation of the velocity, defined respectively on one coarse grid with grid size and one fine grid with grid size and one finite element space for the approximation of the pressure. Nonlinearity and time dependence are both treated on the coarse space. We prove that the difference between the new nonlinear Galerkin method and the standard Galerkin solution is of the order of $H^2$, both in velocity ( and pressure norm). We also discuss a penalized version of our algorithm which enjoys similar properties. Received October 5, 1993 / Revised version received November 29, 1993     

Computation of high order derivatives in optimal shape design

Summary. In shape optimization problems, each computation of the cost function by the finite element method leads to an expensive analysis. The use of the second order derivative can help to reduce the number of analyses. Fujii ([4], [10]) was the first to study this problem. J. Simon [19] gave the second order derivative for the Navier-Stokes problem, and the authors describe in [8], [11], a method which gives an intrinsic expression of the first and second order derivatives on the boundary of the involved domain. In this paper we study higher order derivatives. But one can ask the following questions: -- are they expensive to calculate? -- are they complicated to use? -- are they imprecise? -- are they useless? \medskip\noindent At first sight, the answer seems to be positive, but classical results of V. Strassen [20] and J. Morgenstern [13] tell us that the higher order derivatives are not expensive to calculate, and can be computed automatically. The purpose of this paper is to give an answer to the third question by proving that the higher order derivatives of a function can be computed with the same precision as the function itself. We prove also that the derivatives so computed are equal to the derivatives of the discrete problem (see Diagram 1). We call the discrete problem the finite dimensional problem processed by the computer. This result allows the use of automatic differentiation ([5], [6]), which works only on discrete problems. Furthermore, the computations of Taylor's expansions which are proposed at the end of this paper, could be a partial answer to the last question. Received January 27, 1993/Revised version received July 20, 1993