Frontiers in Mathematical Analysis and Numerical Methods

The following sections are included:

• PREFACE

• CONTENTS

Jacques-Louis Lions was born in the heart of Provence in the charming city of Grasse, much renowned for its perfume industry and its historical center.

In spite of his young age, he had the courage and determination to join the French Résistance at the end of 1943, as a soldier in the FFI (French Forces of the Interior). There he met Andrée, his wife and life-long companion…

On the desk in my office at the Institut Sino-Français de Mathématiques Appliquées (ISFMA), there is a carefully framed photo, which was taken when I saw Professor J.-L. Lions for the last time. That was the afternoon of May 29, 2000, and I was attending the meeting of the (French) Scientific Committee of the ISFMA presided by Prof. J.-L. Lions at the Institut de France in Paris. The participants were in high spirits, and after the meeting a group photo was taken in a quaint hall. I was sitting next to him for that photo and could not imagine that this would be our last time together!…

The authors investigate the stability of a steady ideal plane flow in an arbitrary domain in terms of the L² norm of the vorticity. Linear stability implies nonlinear instability provided the growth rate of the linearized system exceeds the Liapunov exponent of the flow. In contrast, a maximizer of the entropy subject to constant energy and mass is stable. This implies the stability of certain solutions of the mean field equation.

The objective of this paper is to consider the theory of regularity of systems of partial differential equations with Neumann boundary conditions. It complements previous works of the authors for the Dirichlet case. This type of problem is motivated by stochastic differential games. The Neumann case corresponds to stochastic differential equations with reflection on boundary of the domain.

The authors investigate the sensitivity of hydrostatic pressure of flows through porous media with respect to the position of the soil layers. Indeed, these induce discontinuities of the porosity which is a piecewise constant coefficient κ of the partial differential equation satisfied by the pressure u and it leads to the computation of the derivative of u with respect to changes in position of discontinuity surface of κ.

The analysis relies on a mixed formulation of the problem. Preliminary numerical simulations are given to illustrate the theory. An application to a simple inverse problem is also given.

The authors study homogenization of some nonlinear partial differential equations of the form -div (a (hx, h²x, Du_h))= f, where a is periodic in the first two arguments and monotone in the third. In particular the case where a satisfies degenerated structure conditions is studied. It is proved that u_h converges weakly in to the unique solution of a limit problem as h → ∞. Moreover, explicit expressions for the limit problem are obtained.

A simple derivation of the Connection in Finsler space.

This article discusses computational methods for the numerical simulation of unsteady Bingham visco-plastic flow. These methods are based on time-discretization by operator-splitting and take advantage of a characterization of the solutions involving some kind of Lagrange multipliers. The full discretization is achieved by combining the above operator-splitting methods with finite element approximations, the advection being treated by a wave-like equation "equivalent" formulation easier to implement than the method of characteristics or high order upwinding methods. The authors illustrate the methodology discussed in this article with the results of numerical experiments concerning the simulation of wall driven cavity Bingham flow in two dimensions.

This paper is a continuation of the authors' previous paper [1]. In this paper the authors prove, assuming additional conditions on the initial data, some results about the existence and uniqueness of the entropy weak solutions of the Cauchy problem for the singular hyperbolic system

The purpose of the present paper is to call for attention to the following question: Which of the initial data (nonsmall) admit global smooth solutions to the Cauchy problem for nonlinear wave equations. A few cases and examples are sketched, showing that the general answer of this question may be quite complicated.

The authors study the regularity of solutions of the GFD-Stokes problem and of some second order linear elliptic partial differential equations related to the Primitive Equations of the ocean. The present work generalizes the regularity results in [18] by taking into consideration the non-homogeneous boundary conditions and the dependence of solutions on the thickness ε of the domain occupied by the ocean and its varying bottom topography. These regularity results are important tools in the study of the PEs (see e.g. [6]), and they seem also to possess their own interest.

For a class of mixed initial-boundary value problem for general quasilinear hyperbolic systems, this paper establishes the local exact boundary controllability with boundary controls only acting on one end. As an application, the authors show the local exact boundary controllability for a kind of nonlinear vibrating string problem.

Let X denote a complex analytic manifold, and let Aut(X) denote the space of invertible maps of a germ (X,a) to a germ (X,b); this space is obviously a groupoid; roughly speaking, a "Lie groupoid" is a subgroupoid of Aut(X) defined by a system of partial differential equations. To a foliation with singularities on X one attaches such a groupoid, e.g. the smallest one whose Lie algebra contains the vector fields tangent to the foliation. It is called "the Galois groupoid of the foliation". Some examples are considered, for instance foliations of codimension one, and foliations defined by linear differential equations; in this last case one recuperates the usual differential Galois group.

Iterative algorithms for solving the data assimilation problems are considered, based on the main and adjoint equations. Spectral properties of the control operators of the problem are studied, the iterative algorithms are justified.

Several quadrilateral shape regular mesh conditions commonly used in the finite element method are proven to be equivalent. Their influence on the finite element interpolation error and the consistency error committed by nonconforming finite elements are investigated. The effect of the Bi-Section Condition and its extended version (1+α)-Scetion Condition on the degenerate mesh conditions is also checked. The necessity of the Bi-Section Condition in finite elements is underpinned by means of counterexamples.

This paper presents new results for strong solutions and their coincidence sets of the obstacle problem for linear hyperbolic operators of first order. An inequality similar to the Lewy–Stampacchia ones for elliptic and parabolic problems is shown. Under nondegeneracy conditions the stability of the coincidence set is shown with respect to the variation of the data and with respect to approximation by semilinear hyperbolic problems. These results are applied to the asymptotic stability of the evolution problem with respect to the stationary coercive problem with obstacle.

Consider any traveling wave solution of the Kuramoto-Sivashinsky equation that is asymptotic to a constant as x → +∞. The authors prove that it is nonlinearly unstable under H¹ perturbations. The proof is based on a general theorem in Banach spaces asserting that linear instability implies nonlinear instability.

The authors study the asymptotic behaviour of solutions of the heat equation and a number of evolution equations using scaling techniques. It is proved that in the framework of bounded data stabilization need not occur and the general asymptotic behaviour is complex. This behaviour reflects for large times, even on compact sets, the complexity of the initial data at infinity.