Remarks on the null controllability of the Burgers equation

In the context of the Burgers equation with distributed controls, we present optimal estimates for the minimal time of controllability $T(r)$ of the initial data of norm ?r in $L^2$. To cite this article: E. Fernández-Cara, S. Guerrero, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Controllability of a multichannel system

We consider the system consisting of K coupled acoustic channels with the different sound velocities $c_j$. Channels are interacting at any point via the pressure and its time derivatives. Using the moment approach and the theory of exponential families with vector coefficients we establish two controllability results: the system is exactly controllable if(i) the control $u_j$ in the jth channel acts longer than the double travel time of a wave from the start to the end of the j-th channel;(ii) all controls $u_j$ act more than or equal to the maximal double travel time.     

Superefficient drift estimation on the Wiener space

In the framework of a nonparametric functional estimation for the drift of a Brownian motion $X_t$ we construct Stein type estimators of the form $X_t+D_t\log F$ which are superefficient when $\sqrt{F}$ is a superharmonic functional on the Wiener space for the Malliavin derivative D. To cite this article: N. Privault, A. Réveillac, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

p-adic affine dynamical systems and applications

We consider p-adic affine dynamical systems on the ring ${\mathbb{Z}}_p$ of all p-adic integers, and we find a necessary and sufficient condition for such a system to be minimal. The minimality is equivalent to the transitivity, the ergodicity of the Haar measure, the unique ergodicity, and the strict ergodicity. When the condition is not satisfied, we prove that the system can be decomposed into strict ergodic subsystems. One of our applications is the study of the divisibility, by a power of prime number, of the sequence of integers $a^n?b$ with positive integers $a,b$ and n. To cite this article: A.-H. Fan et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Uniform asymptotic formulae for Green's kernels in regularly and singularly perturbed domains

Asymptotic formulae for Green's kernels $G_{\epsilon }(\mathbf{x},\mathbf{y})$ of various boundary value problems for the Laplace operator are obtained in regularly perturbed domains and certain domains with small singular perturbations of the boundary, as $\epsilon \to 0$. The main new feature of these asymptotic formulae is their uniformity with respect to the independent variables x and y. To cite this article: V. Maz'ya, A. Movchan, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Résolutions de singularités non-abéliennes en dimension trois

Let E be an elliptic curve and R be a real three dimensional root system. Let W be the Weyl group associated to R. Denote $W_{+}=W\cap {\mathrm{SL}}_3(\mathbb{C})$. The quotient $E?Q(\mathrm{R})/W_{+}$ admits two natural crepant resolutions. One is the result of a Jung process of desingularization of singularities, the other the equivariant Hilbert scheme. To compare these resolutions, we calculate the fibres over any singular point in both cases. We exhibit a McKay correspondence phenomenon for the Hilbert scheme resolution and construct a new family of vector bundles on E parametrized by the $W_{+}$-Hilbert scheme. To cite this article: S. Térouanne, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

An isoperimetric inequality for the principal eigenvalue of the Laplacian with drift

We generalize the classical Rayleigh–Faber–Krahn inequality to the case of the Dirichlet Laplacian with a drift. We also solve some optimization problems for the principal eigenvalue of the operator $?\mathrm{\Delta }+v?\mathrm{?}$ in a fixed domain with a control of the drift v in $L^{\infty }$. To cite this article: F. Hamel et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Critical space for the parabolic-parabolic Keller–Segel model in Rd

We study the Keller–Segel system in ${\mathbb{R}}^d$ when the chemoattractant concentration is described by a parabolic equation. We prove that the critical space, with some similarity to the elliptic case, is that the initial bacteria density satisfies $n_0\in L^a({\mathbb{R}}^d)$, $a>d/2$, and that the chemoattractant concentration satisfies $\mathrm{?}{c}_0\in L^d({\mathbb{R}}^d)$. In these spaces, we prove that small initial data give rise to global solutions that vanish as the heat equation for large times and that exhibit a regularizing effect of hypercontractivity type. To cite this article: L. Corrias, B. Perthame, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

A class of nonpositively curved Kähler manifolds biholomorphic to the unit ball in Cn

Let $(M,g)$ be a simply connected complete Kähler manifold with nonpositive sectional curvature. Assume that g has constant negative holomorphic sectional curvature outside a compact set. We prove that M is then biholomorphic to the unit ball in ${\mathbb{C}}^n$, where ${\dim }_{\mathbb{C}}M=n$. To cite this article: H. Seshadri, K. Verma, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Piatetski-Shapiro phenomenon in the uniqueness problem

We extend the phenomenon discovered by Piatetski-Shapiro (1954) to $l^q$ spaces. To be precise, for any $q>2$ we construct a compact K on the circle, which supports a distribution S with Fourier transform $\stackrel{?}{S}\in l^q$, but does not support such a measure. To cite this article: N. Lev, A. Olevskii, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Analyse anaplectique sur la droite réelle

The two irreducible summands of the metaplectic representation in $L^2(\mathbb{R})$ occur for one in the discrete series of the twofold cover of the group $G=\mathrm{SL}(2,\mathbb{R})$, for the other in the continuation of this series. Starting instead from the complementary series of the universal cover of G, one can associate with every real number ν mod 2 a new ‘analysis’ on the real line. The space ${\mathfrak{A}}_{\nu }$ that substitutes for $L^2(\mathbb{R})$ consists of functions extending as entire functions of one variable. Once the proper notion of integral on the real line has been introduced, one easily defines a concept of Fourier transformation and a new ‘ν-anaplectic’ representation: the latter combines in the usual way with the Heisenberg representation, and both representations are pseudo-unitary with respect to some ν-dependent scalar product. The spectrum of the harmonic oscillator in the space ${\mathfrak{A}}_{\nu }$ is the set $\nu +\frac{1}{2}+\mathbb{Z}$. To cite this article: A. Unterberger, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Monotone approximations of Green's functions

We study the approximations of the Green's function $\mathbb{G}$ in a domain Ω obtained from an approximation of the Dirac mass ${\delta }_0$. We prove that under some conditions, these approximations converge monotonically to $\mathbb{G}$, a rather surprising result. To cite this article: E. Chasseigne, R. Ferreira, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

A unified fictitious domain model for general embedded boundary conditions

This Note addresses the analysis of a new fictitious domain method for elliptic problems in order to handle general embedded boundary conditions (E.B.C.): Fourier, Neumann and Dirichlet conditions on an immersed interface. Our method is based on a recent model of fracture combining flux and solution jumps on the interface Σ separating the original domain $\stackrel{?}{\Omega }$ from the auxiliary exterior domain ${\Omega }_e$. A class of methods is derived within the same unified formulation with either no penalty or exterior control in ${\Omega }_e$, or surface penalty on Σ, volume $H^1$ or $L^2$ penalty in ${\Omega }_e$, or both. The consistency (no penalty) or optimal error estimates with respect to the penalty parameter are proved for such methods. To cite this article: Ph. Angot, C. R. Acad. Sci. Paris, Ser. I 341 (2005).