We give the definitions of exact and approximate controllability for linear and nonlinear Schrödinger equations, review fundamental criteria for controllability and revisit a classical “No-go” result for evolution equations due to Ball, Marsden and Slemrod. In Section 2 we prove corresponding results on non-controllability for the linear Schrödinger equation and distributed additive control, and we show that the Hartree equation of quantum chemistry with bilinear control $\left(E\right(t)·x)u$ is not controllable...

Dans cette article, on étudie la limite lorsque m --> ∞ de la solution du problème de Cauchy ut - ∆um + div F(u) = 0 sur un ouvert Omega avec des conditions aux bords de type Dirichlet et une donnée initiale u0 ≥ 0.

Les effets dispersifs permettent de passer à la limite dans le système d’Euler compressible 2-D isentropique, quand le nombre de Mach tend vers zéro, même si les données initiales ne sont pas uniformément régulières.Ceci mène à des résultats de convergence vers des solutions non régulières du système d’Euler incompressible, comme les poches de tourbillon ou les solutions de Yudovich.

L’objet de cette note est d’étudier la limite quasineutre des équations de Vlasov Poisson en dimension $1$ d’espace. Ceci inclut l’obtention de résultats d’existence pour le système limite ainsi que la preuve de la convergence.

Il s’agit de comparer les différents résultats et théorèmes concernant dans un cadre essentiellement déterministe des systèmes de particules. Cela conduit à étudier la notion de hiérarchies d’équations et à comparer les modèles non linéaires et linéaires. Dans ce dernier cas on met en évidence le rôle de l’aléatoire. Ce texte réfère à une série de travaux en collaboration avec F. Golse, A. Gottlieb, D. Levermore et N. Mauser.

The behavior of an ordinary differential equation for the low wave number velocity mode is analyzed. This equation was derived in [5] by an iterative process on the two-dimensional Navier-Stokes equations (NSE). It resembles the NSE in form, except that the kinematic viscosity is replaced by an iterated viscosity which is a partial sum, dependent on the low-mode velocity. The convergence of this sum as the number of iterations is taken to be arbitrarily large is explored. This leads to a limiting...

The limiting behavior of global attractors ${𝒜}_{\epsilon }$ for singularly perturbed beam equations $${\epsilon }^2\frac{{\partial }^{2}u}{\partial t^2}+\epsilon \delta \frac{\partial u}{\partial t}+A\frac{\partial u}{\partial t}+\alpha Au+{g(\parallel u\parallel }_{1/4}^2)A^{1/2}u=0$$ is investigated. It is shown that for any neighborhood $𝒰$ of ${𝒜}_0$ the set ${𝒜}_{\epsilon }$ is included in $𝒰$ for $\epsilon$ small.

The estimate ${∥D^{k-1}u∥}_{L^{n/(n-1)}}\le {∥A\left(D\right)u∥}_{L^1}$ is shown to hold if and only if $A\left(D\right)$ is elliptic and canceling. Here $A\left(D\right)$ is a homogeneous linear differential operator $A\left(D\right)$ of order $k$ on ${ℝ}^n$ from a vector space $V$ to a vector space $E$. The operator $A\left(D\right)$ is defined to be canceling if ${\bigcap }_{\xi \in {ℝ}^n\setminus \left\{0\right\}}A\left(\xi \right)\left[V\right]=\left\{0\right\}$. This result implies in particular the classical Gagliardo–Nirenberg–Sobolev inequality, the Korn–Sobolev inequality and Hodge–Sobolev estimates for differential forms due to J. Bourgain and H. Brezis. In the proof, the class of cocanceling homogeneous linear differential...

The nonlocal boundary value problems for linear and nonlinear degenerate abstract differential equations of arbitrary order are studied. The equations have the variable coefficients and small parameters in principal part. The separability properties for linear problem, sharp coercive estimates for resolvent, discreetness of spectrum and completeness of root elements of the corresponding differential operator are obtained. Moreover, optimal regularity properties for nonlinear problem is established....