Jacques-Louis Lions

Mathematical Analysis and Numerical Methods for Science and Technology

These six volumes - the result of a ten year collaboration between the authors, two of Frances leading scientists and both distinguished international figures - compile the mathematical knowledge required by researchers in mechanics, physics, engineering, chemistry and other branches of application of mathematics for the theoretical and numerical resolution of physical models on computers. Since the publication in 1924 of the Methoden der mathematischen Physik by Courant and Hilbert, there has been no other comprehensive and up-to-date publication presenting the mathematical tools needed in applications of mathematics in directly implementable form. The advent of large computers has in the meantime revolutionised methods of computation and made this gap in the literature intolerable: the objective of the present work is to fill just this gap. Many phenomena in physical mathematics may be modeled by a system of partial differential equations in distributed systems: a model here means a set of equations, which together with given boundary data and, if the phenomenon is evolving in time, initial data, defines the system. The advent of high-speed computers has made it possible for the first time to caluclate values from models accurately and rapidly. Researchers and engineers thus have a crucial means of using numerical results to modify and adapt arguments and experiments along the way. Every fact of technical and industrial activity has been affected by these developments. Modeling by distributed systems now also supports work in many areas of physics (plasmas, new materials, astrophysics, geophysics), chemistry and mechanics and is finding increasing use in the life sciences. Volumes 5 and 6 cover problems of Transport and Evolution.

Exact controllability, stabilization and perturbations for distributed systems

Exact controllability is studied for distributed systems, of hyperbolic type or for Petrowsky systems (like plate equations).The control is a boundary control or a local distributed control. Exact controllability consists in trying to drive the system to rest in a given finite time. The solution of the problems depends on the function spaces where the initial data are taken, and also depends on the function space where the control can be chosen.A systematic method (named HUM, for Hilbert Uniqueness Method) is introduced. As the terminology indicates, it is based on Uniqueness results (classical or new) and on Hilbert spaces constructed (in infinitely many ways) by using Uniqueness. A number of applications are indicated.Having a general method for exact controllability implies having a general method for stabilization. This leads to new (and complicated ...) nonlinear Riccati’s type PDEs, to be compared with direct methods (when available).In the last part of the paper, we consider how all this behaves fo...

On Some Questions in Boundary Value Problems of Mathematical Physics

Publisher Summary This chapter introduces some questions that arise in boundary value problems of mathematical physics. Some problems of the hydrodynamics of incompressible nonhomogeneous fluids are described in the chapter. The chapter describes the equations of flows of incompressible fluids that are nonhomogeneous in the sense of not having a constant density. The classical Navier–Stokes equations are described in the chapter. Except in some details of presentations, the chapter follows the notes of Antonzev and Kajikov. The problems studied in Antonzev and Kajikov include in particular the existence of strong solutions in the two dimensional case. Statement of existence theorem is presented and Galerkins approximation is discussed in the chapter. A discussion is presented in the chapter on a linear equation arising in the kinetic theory of gases and containing some nonstandard aspects. An introduction to the method of homogenization for composite materials is also given in the chapter.

REITERATED HOMOGENIZATION OF NONLINEAR MONOTONE OPERATORS

In this paper, the authors study reiterated homogenization of nonlinear equations of the form -div(a(x,x/e,x/e2,Due))=f, where a is periodic in the first two arguments and monotone in the third. It is proved that ue converges weakly in W1,p(Ω) (and even in some multiscale sense), as e→0 to the solution u0 of a limit problem. Moreover, an explicit expression for the limit problem is given. The main results were also stated in [15]. This article presents the complete proofs of these results.

On exact and approximate boundary controllabilities for the heat equation: a numerical approach

The present article is concerned with the numerical implementation of the Hilbert uniqueness method for solving exact and approximate boundary controllability problems for the heat equation. Using convex duality, we reduce the solution of the boundary control problems to the solution of identification problems for the initial data of an adjoint heat equation. To solve these identification problems, we use a combination of finite difference methods for the time discretization, finite element methods for the space discretization, and of conjugate gradient and operator splitting methods for the iterative solution of the discrete control problems. We apply then the above methodology to the solution of exact and approximate boundary controllability test problems in two space dimensions. The numerical results validate the methods discussed in this article and clearly show the computational advantage of using second-order accurate time discretization methods to approximate the control problems.

Heterogeneous coupling by virtual control methods

Summary. The virtual control method, recently introduced to approximate elliptic and parabolic problems by overlapping domain decompositions (see [7–9]), is proposed here for heterogeneous problems. Precisely, we address the coupling of an advection equation with a diffusion-advection equation, with the aim of modelling boundary layers. We investigate both overlapping and non-overlapping (disjoint) subdomain decompositions. In the latter case, several cost functions are considered and a numerical assessment of our theoretical conclusions is carried out.

Domain decomposition methods for CAD

Abstract Constructive solid geometry (CSG) in CAD leads to Domain decompositions which are based on primitive shapes, as briefly explained in the introduction below. A special role is played by “holes”, which can be viewed in several different ways. We combine this remark with the method of s. In a previous. In a previous note [7] where this method was introduced, the distributions resulting from the decomposition and Greens formula where thought of as virtual controls — here the virtual controls are introduced a priori with s, or itoutside the, or e the do the domain, or in the intersections of the domains. They are then chosen so as to (approximately) satisfy the Boundary conditions, which is possible by virtue of s (which have to be proven!). These (which have to be proven!). These ideas are presented here on an example (Section 1) which is certainly not the most general one but which is sufficient to show how everything extends to very many other situations. Algorithms, following the Approximate controllability lemma, are given in Section 2 and a numerical example is presented in Section 4.

Algorithmes parallèles pour la solution de problèmes aux limites

We present in this Note a number of rather systematic methods for constructing parallel algorithms for the approximation of the solution of Boundary Value Problems with the aim of arriving at methods which are hopefully valid for “all” boundary value problems, linear or not, of any order. These methods are based on domain decomposition, but in fact, as it will be shown in a separated paper, they can apply as well to frequency decompositions. We use here the notations introduced in [4], where the goal was to find (parallel) algorithms of stabilization. The main goal here is the parallel solution of optimal control problems, a summary of this being presented in the next note [5]. The tools introduced for the control problems may have some interest for the solution of boundary value problems, without control. This is presented here. After a brief introduction (of the “abstract” type) in Section 1, several examples are given in Section 2. Several other situations, using similar principles to those introduced here, will be given in separated papers (already in [6]). There is a huge litterature devoted to problems of the type of those introduced here. The idea used in Example 3 has already been introduced (and used in a different way) by Yu. Kuznetsov [2], T. Mathew et al. [7] and P. LeTallec et al. [3].

A Computational Approach to Controllability Issues for Flow-Related Models. (I): Pointwise Control of the Viscous Burgers Equation

We discuss the numerical solution of some controllability problems for time-dependent flow models. The emphasis is on algorithmic aspects, discretization issues, and memory-saving devices. In the first part of the article, we investigate the controllability of the viscous Burgers equation. In part two, we shall discuss the boundary controllability of a linear advection-diffusion equation and then the distributed controllability of the unsteady Stokes equations.

Well-posed absorbing layer for hyperbolic problems

Summary. The perfectly matched layer (PML) is an efficient tool to simulate propagation phenomena in free space on unbounded domain. In this paper we consider a new type of absorbing layer for Maxwells equations and the linearized Euler equations which is also valid for several classes of first order hyperbolic systems. The definition of this layer appears as a slight modification of the PML technique. We show that the associated Cauchy problem is well-posed in suitable spaces. This theory is finally illustrated by some numerical results. It must be underlined that the discretization of this layer leads to a new discretization of the classical PML formulation.