Remarks on the controllability of some parabolic equations and systems

We present a review of recent results concerning the null controllability of parabolic systems. Among others, we will consider the heat equation, the Burgers, Navier-Stokes and Ginzburg-Landau equations, etc. We will also indicate some open questions.Dirección General de Investigació

A review of basic theoretical results concerning the Navier-Stokes and other similar equations

These notes are devoted to provide an introductory approach to the Navier-Stokes and some other related equations. Most concepts and arguments recalled below are very general and we believe that this presentation can be of help for the theoretical analysis of many PDE’s arising from Sciences and Engineering. First, we recall the Navier-Stokes equations, we explain the meaning of the variables and data and we state some technical results needed for our study. Then, we state and give the proofs of some basic existence, uniqueness and regularity results. In the proof of existence, we apply usual compactness arguments to a family of Galerkin approximations. We also discuss briefly some of the main open problems arising in the three-dimensional case. In a final section, we review briefly the state of the art for other similar equations and we indicate some related open questions

Null controllability of the semilinear heat equation

This paper is concerned with the null controllability of systems governed by semilinear parabolic equations. The control is exerted either on a small subdomain or on a portion of the boundary. We prove that the system is null controllable when the nonlinear term f(s) grows slower than s . log|s| as |s| → ∞.Ce papier est consacré à l'étude de la contrôlabilité nulle des systèmes gouvernés par des équations aux dérivées partielles paraboliques semilinéaires. Le contrôle est localisé dans un (petit) sous-domaine ou sur une (petite) partie de la frontière. Les résultats principaux montrent que le système est exactement contrôlable à zéro si le terme non linéaire f(s) croît plus lentement que s . log|s| quand |s| tend vers l'infini.Dirección General de Investigación Científica y Técnic

Several questions concerning the control of parabolic systems

This paper is devoted to recall several recent results concerning the null controllability of some parabolic systems. Among others, we will consider the classical heat equation, the Burgers, Navier-Stokes and GinzburgLandau equations, etc.Dirección General de Investigación Científica y Técnic

On the approximate and null controllability of the Navier-Stokes equations

This paper presents some known results on the approximate and null controllability of the Navier–Stokes equations. All of them can be viewed as partial answers to a conjecture of J.-L. Lions.Dirección General de Investigación Científica y Técnica

Motivation, analysis and control of the variable density Navier-Stokes equations

Résultats d’unicité pour des problèmes pseudomonotones avec p > 2. Nous considérons un opérateur pseudomonotone du type −div(b(x, u)|∇ u| p−2∇ u), avec 1 2. Les démonstrations complètes sont données dans cette Note. Les résultats restent valides quand Ω est non borné